고유 소수

고유 소수
No. 알려진.	102
용어의 추측	무한
제1항	3, 11, 37, 101
가장 큰 알려진 용어	(10270343-1)/9
OEIS 지수	A040017; 고유 주기 프리타임(다른 프라이머리는 1/p와 같은 기간을 가지지 않음) 순서(페어리는 A051627에 제공됨);

레크리에이션 수 이론에서, 독특한 전성기 또는 독특한 기간의 전성기는 특정한 종류의 프라임 수이다.전형적인 p≠ 2가 없는 경우 다른 주요 q등은 상호, 1/p의 소수 전개의 기간 길이 q의 역수의 기간 길이와 같다는, 기간 1로 예를 들어 1/q.[1]3은 유일한 전성기, 기간 2에서 11명이 있는 유일한 전성기, 기간 3가 37은 유일한 전성기, 101은 유일한 주요 지..지 5독특한이라고 불린다그것4교시야, 그래서 그들은 독특한 프라임이야.In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both18세, 3541세, 27961세 둘 다 20세기를 갖는다.따라서 이 중 어느 것도 독특한 전성기는 아니다.독특한 프라임은 1980년에 사무엘 예이츠에 의해 처음 묘사되었다.

위의 정의는 정수의 소수점 표시와 관련이 있다.고유한 소수점 이하에서 정의될 수 있으며 연구되어 왔다.

b 기준의 프라임 기간

숫자 $b$ 에서 prime 숫자(또는 보다 일반적으로 정수) $p$ 의 역수(recouncil)는 다음과 같은 경우 주기 $n$ 이다.

{\displaystyle {\frac {1}{p1}=\sum _{i=1}^{\inflt }{\frac {q}{(b^{n}}}^{i}}}}}}}}},

여기서 q는 b $b^{n}.$ . $b^{n$ 보다 작은 양의 정수다. $}}$ 기하계열의 합계식에 의하면 $b^{n}.$ , 이것을 다음과 같이 다시 쓸 수도 있다.

{\frac {1}{p}={\frac {q}{b^{n}-1}.

In other words, $n$ is a period of the representation of $1/ p$ if and only if $p$ is a divisor of $b^{n}-1.$ Euler's theorem asserts that, if an integer $b$ is coprime with $p$ , then $p$ is a divisor of $b^{\varphi (p)}-1$ where $\varphi$ is Euler's totient func이것은 $p$ 를 가진 모든 $정수$ b coprime에 대해 $p$ 의 역수 표시가 base $b$ 에서 주기적이라는 것을 증명한다.

주기함수의 모든 기간은 일반적으로 기본기라 불리는 최단기의 배수다.이 글에서 우리는 $base$ b에서 $p$ 의 기간을 base $b$ 에서1/ $p$ 를 나타내는 가장 짧은 기간이라고 부른다.따라서 base $b$ 에서 $p$ 의 기간은 $b^{n}-1.$ 가 b n $b^{n}-1.$ - $.[\displaystyle b^{n}-1.}$ 의 구분자일 정도로 가장 작은 양의 $정수$ n이다. 즉 $b^{n}-1.$ , base $b$ 에서 prime $p$ 의 기간은 b $modulo$ $p$ 의 승수 순서가 된다.

지그몬디의 정리에 따르면, 모든 양의 정수는 다음과 같은 경우를 제외하고는 베이스 $b$ 에서 어느 정도 전성기가 되는 기간이다.

$b$ = $2$ 및 n = 1 $또는$ 6
$n$ = $2$ 및 $b = 일부 k 정수$ $k$ > $1$ 의 경우 2 - 1

로서

x^{n}-1=\prod _{i\mid n}\Phi _{n}(x),

여기서 $\Phi _{n}$ ${\$ 는 $\Phi _{n}$ n번째 사이클로토믹 다항식이며, b에서 $주기$ $n$ 의 소수점은 $\Phi _{n}(b).$ ( $\Phi _{n}(b).$ ) $\Phi _{n}(b).$ . ${\displaystyle \Phi _{n}(b$ )의 소수점이다 $.$ $}$ 더 정확히 $\Phi _{n}(b).$ 말하면, 기간 $n$ 의 소수점은 $정확히$ n을 나누지 않는 $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ 의 주요 구분점이다(이 결과 및 다음 결과의 증거는 아래 참조).

$b$ 가 짝수(이진수 및 소수점 이하 포함)인 경우, $n$ 을 나누지 $\Phi _{n}(b)$ 않는 $\Phi _{n}(b)$ $\Phi _{n}(b)$ ) $\Phi _{n}(b)$ {\ $displaystyle \Phi _{n}(b)}$ 의 소수점(primary divisors)은 정확히^{[citation needed]} 의 소수점이다.

R_{n}(b)={\frac {\frac _{n}(b)}{\gcd(\Phi _{n})(b,n)}}}}

$b$ 가 홀수인 경우 이는 잘못된 것이다: n= $2$ 및 b = $4k$ - $1$ 인 경우, 여기서 $k$ 는 양의 정수인 경우,

R_{2}(b)={\frac {\frac _{2}(b)}{\gcd(\Phi _{2}(b),2}}={\frac {b+1}{2}}=2k,

2는 $n$ = $2$ 와 $\Phi _{n}(b).$ ( $\Phi _{n}(b).$ )을 모두 나눈다 $\Phi _{n}(b).$ ${\displaystyle \Phi _{n}(b)$ $}$

$b$ 가 홀수일 경우 기간 $n$ 의 소수 n은 정확히, $n$ = $1$ 이면 R $R_{1}(b)=b-1$ ( $R_{1}(b)=b-1$ = $R_{1}(b)=b-1$ - $R_{1}(b)=b-1$ {\ $displaystyle R_{1}(b)=b-1$ $n$ > $1$ 이면 $R n (b$ )의 홀수 소수점이다 $.$

PRIME의 특성화에 대한 증거의 스케치 n

모든 prime $p$ 의 기간이 $p$ – $1$ (페르마트의 작은 정리)을 나누기 $때문$ 에 $p$ 가 n을 나누면 그 기간이 $n$ 보다 작다.반대로 $p$ 가 $\Phi _{n}(b)$ ( $\Phi _{n}(b)$ ) ${\$ $displaystyle$ $\Phi _{n}(b)}$ 을(를) 나누고 $\Phi _{n}(b)$ 마침표 $k$ 를 $n$ 보다 작다면, $φn$ $)과$ $\Phi _{k}(b).$ $)$ 의 공통점이다 $.$ $}}$ 두 다항식의 결과물이 이들 다항식의 선형 결합이므로 $\Phi _{k}(b).$ $p$ 는 $\Phi _{n}(x)$ ( $\Phi _{n}(x)$ ) ${\displaystyle \Phi _{n}($ $\Phi _{k}(x).$ $)$ 와 $\Phi _{n}(x)$ $\Phi _{k}(x).$ $\Phi _{k}(x).$ ( x $\Phi _{k}(x).$ ) $\Phi _{k}(x).$ . ${\displaystyle \Phi _{k}(x$ )의 결과물을 나눈다 $.$ $}$ As these two polynomials are coprime and divide $x^{n}-1,$ $p$ divides also the discriminant $n^{n}$ of $x^{n}-1.$ Thus, a prime divisor of $\Phi _{n}(b)$ , that has a period smaller than $n$ , is also a divisor of $n$ .

이제 프라임 $p$ > $2가$ $\Phi _{n}(b)/p.$ 과 $\Phi _{n}(b),$ ( $\Phi _{n}(b),$ $\Phi _{n}(b),$ , ${\displaystyle \Phi _{n}(b)$ 를 $\Phi _{n}(b),$ 나누면 dividesn ( $\Phi _{n}(b)/p.$ ) $\Phi _{n}(b)/p.$ / $\Phi _{n}(b)/p.$ $\Phi _{n}(b)/p.$ $displaystyle \Phi _{n}(b)/p$ 를 나누지 않는다는 것을 증명해야 한다 $.$ $}}$ 사실 $\Phi _{n}(b)/p.$ 이는 p가 $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ ( b $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ ) $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ / $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ ( $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ ( b $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ ) $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ , $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ . ${\displaystyle \Phi \{n}(b)/\gcd(\Phi _{n})(b,n)$ 를 나누지 않는다는 것을 즉시 암시한다 $.$ $}$ $b$ 가 짝수인 경우 $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ 2는 $\Phi _{n}(b),$ ( $\Phi _{n}(b),$ ) $\Phi _{n}(b),$ , ${\displaystyle \Phi _{n}(b),}($ 이상한 경우)를 나눌 수 $없으며$ p > 2 조건도 제한적이지 않다.

$따라서$ n = $pm$ 으로 한다. $p^{2}$ $p^{2}$ ${\$ p $^{2}}:$ 일부 다항식 $S (x$ )에 대해 $S (b)$ 를 나누지 않는다는 $p^{2}$ 것을 증명하는 것으로 충분하다 $.$ 이 S(b)는 $\Phi _{n}(x).$ ( $\Phi _{n}(x).$ $\Phi _{n}(x).$ . ${\displaystyle \Phi _{n}(x$ )의 배수다 $.$ ${$ 우리가 가져갈게 $\Phi _{n}(x).$

S(x)={\frac {x^{n}-1}{x^{m}-1}=1+x^{m}+x^{2m}+\cdots +x^{(p-1)m}.

By Fermat's little theorem, we have $b^{p-1}\equiv 1{\pmod {p}}.$ As $p$ divides $\Phi _{n}(b)$ , we have also $b^{n}\equiv 1{\pmod {p}}.$ Thus the multiplicative order of $b$ modulo $p$ divides $gcd(n, p - 1)$ , which i $s$ m = $n / p$ 의 구분자.따라서 c = b - $1$ 은^m $p$ 의 배수다.지금

S(b)={\frac{1+c)^{p-1}{c}{p}{p}{p}{p}{p}{p}}{p}}}{p-1}.

$p$ 가 prime이고 2보다 크므로 첫 번째 용어를 제외한 모든 용어는 p $p^{2}.$ . ${\displaystyle$ p $^{2$ }의 배수임 $.$ $p^{2}.$ 이것은 $p^{2}$ ${\$ p $^{2$ }}개가 $p^{2}$ $\Phi _{n}(b).$ $\Phi _{n}(b).$ $\Phi _{n}(b).$ . ${\displaystyle \Phi _{n}(b$ )를 나누지 않는다는 $p^{2}$ 것을 증명한다 $.$ $}$

prime $p$ 는 base $b$ 에서 고유한 prime이며, 일부 $n$ 의 경우 $n$ 을 나누지 않는 것이 $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ 의 고유한 primeisor인 경우에만 해당된다. $b$ 가 짝수(이진수 및 소수점 이하 포함)라면 이는 다음을 의미한다.

R_{n}(b)={\frac {\frac _{n}(b)}{\gcd(\Phi _{n})(b,n)}}}}}=p^{c}.

일부 양의 $정수$ c .

$b$ 가 홀수라면 이는

R_{n}(b)={\frac {\frac _{n}(b)}{\gcd(\Phi _{n}b,n)}}}}}}=p^{c}2^{d}.

$일부$ 정수 c > $0$ 및 $d$ ≥ $0$ 의 경우.이것은 주어진 기간의 고유한 소수점과 소수점을 계산하는 효율적인 방법을 제공한다.

$b$ 의 주요 구분자는 b $b^{n}-1$ - $b^{n}-1$ ${\displaystyle$ b $^{n}-1$ 과 동일하므로, 구분자 $\Phi _{n}(b).$ ( $\Phi _{n}(b).$ ) $\Phi _{n}(b).$ . $\Phi _{n}(b).$ {\ $displaystyle \Phi _{n}(b$ )과도 동일하다는 점에 유의하십시오 $.$ $}}$ 그러한 $\Phi _{n}(b).$ 프라임은 상호의 b $기저$ 에서의 표현은 주기적인 것이 아니라 유한하기 때문에 주기적인 길이가 없다.따라서 그러한 프라임은 베이스 $b$ 에서 유한한 호혜를 갖는 고유한 프라임이라 하더라도 결코 독특한 프라임으로 간주되지 않는다.예를 들어, 2는 2진수에서 유한한 역수를 갖는 유일한 프라임이지만 2진수에서는 고유한 프라임으로 간주되지 않는다.

base $b$ 에는 고유한 전성기의 네 가지 상황이 있다.

$\Phi _{n}(b)=p$ ( $\Phi _{n}(b)=p$ ) $\Phi _{n}(b)=p$ = $\Phi _{n}(b)=p$ ${\displaystyle \Phi _{n}(b)=p},$ $p$ 는 $\Phi _{n}(b)=p$ prime이다.
${\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p$ and $p$ is prime and $\gcd(\Phi _{n}(b),n)>1$ .
$\Phi _{n}(b)=p^{c}$ ( $\Phi _{n}(b)=p^{c}$ ) $\Phi _{n}(b)=p^{c}$ = $\Phi _{n}(b)=p^{c}$ $\Phi _{n}(b)=p^{c}$ {\ $displaystyle \Phi _{n}(b)=p^{c}},$ $p$ 는 $\Phi _{n}(b)=p^{c}$ prime이고 $c>1$ > 1 ${\displaystyc>1$
${\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p^{c}$ and $p$ is prime and $\gcd(\Phi _{n}(b),n)>1$ and $c>1$ .

추측:

모든 양의 정수 n에 대해, 기간 길이 n을 가진 고유한 전성기가 있을 정도로 무한히 많은 염기 b가 있다. (이 추정은 사이클로토모스 다항식은 해석할 수 없기 때문에, Bunyakowsky 추측이 사실이라면 사실이다.)

모든 베이스 b에 대해, 무한히 많은 독특한 프라임이 있다.

모든 양의 정수 n에 대해, 첫 번째 상황인 기간 길이 n을 가진 고유한 prime이 있을 정도로 base가 무한히 많다.(i.e. for all n≥1, there are infinitely many b≥2 such that $\Phi _{n}(b)$ is prime, where $\Phi$ is the cyclotomic polynomial) (this conjecture is true if Bunyakowsky conjecture is true, since cyclotomic polynomials are irreducible)

모든 베이스 b에 대해, 첫 번째 상황인 독특한 프리임이 무한히 많다.(즉, 모든 b≥2에 대해, $\Phi _{n}(b)$ ( $\Phi _{n}(b)$ $\Phi _{n}(b)$ $\Phi _{n}(b)$ 이 $\Phi _{n}(b)$ (가) prime이고, $\Phi$ 서 φ ${\displaystyle$ \ $Phi }$ 은 $\Phi$ 사이클로토믹 다항식)과 같이 n≥1이 무한히 많다.

n이 아닌 모든 양의 정수(OEIS의 순서 A253235)에 대해, 두 번째 상황인 기간 길이 n을 가진 고유한 prime이 있을 정도로 b가 무한히 많다.(주기 길이 n in (OEIS의 순서 A253235)은 2번째 또는 4번째 상황인 베이스를 가질 수 없음) (이 추측은 사이클로토모믹 다항식(cyclotomic polyomials)이 불가해하므로 Bunyakowsky 추측이 사실이라면 사실이다)

모든 베이스 b의 경우, 2번째 상황인 고유 프리타임(또는 고유한 프리타임 없음)이 미세하게 많을 뿐이며, 특히 베이스 2에서는 2번째 상황인 것으로 알려진 기간이 18, 20, 21, 54, 147, 342, 602, 889, 258121이며 베이스 10에서는 유일하게 3, 9, 294이다.

세 번째 상황인 모든 (주기 n, b) 콤보는

- n = 1, b = p p prime 및 c ≥ 2인 p^c+1
- n = 2, b = p 홀수 prime 및 c ≥ 2인 p-1^c
- n = 3, b = 18
- n = 5, b = 3
- n = 6, b = 19

4번째 상황인 모든 (주기 n, b) 콤보는

- n = 2, b = 2p-1^r^c p 홀수 prime 및 r ≥ 1 및 c ≥ 2
- n = 3, b in (b²+b+1)/3의 경우 (OEIS의 경우 시퀀스 A028231)는 프라임의 제곱(지수가 2 이상인 힘은 존재할 수 없다고 추측된다) (첫 번째 베이스 b는 22, 313, 4366, 847078, ...) (상속 프라임이 (OEIS의 경우)에 있다) (상속).
- n = 4, b in (sequence A002315 in the OEIS) in the case of (b²+1)/2 is square of prime or 28561=13⁴ (it is conjectured that powers with exponent > 2 cannot exist, except (239²+1)/2 = 28561 = 13⁴) (the first such bases b are 7, 41, 239, 8119, 47321, ...) (correspond prime are in (sequence A001563 in the OEIS))
- n = 6, b-1 in (b-b²+1)/3의 경우 (OEIS의 경우 후속 A028231)은 프라임의 제곱(지수가 2 이상인 힘은 존재할 수 없다고 추측된다) (첫 번째 그러한 베이스 b는 (23, 314, 4367, 847079, ...) (상속 프라임은 (시퀀스 A001570 in OEIS)

특히 n > 6에 대해서는 3번째 또는 4번째 상황인 no(주기 n, b) 콤보가 존재한다.

"거의 다" (주기 n, 베이스 b) 콤보가 첫 번째 상황이다.

(3번째 또는 4번째 상황인 모든 (주기 n, base b) 콤보에 대해 해당 prime p는 반드시 Wieferich prime to base)를 일반화해야 한다.

최대 24개의 기준에서 139까지의 프라임 기간 표

"종단"이라는 말은 프라임이 베이스를 나누는 것을 의미하며, 따라서 그 상호의 표현은 유한하다는 것을 의미한다.

밑의 전성기의	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨	1	종료됨
3	2	종료됨	1	2	종료됨	1	2	종료됨	1	2	종료됨	1	2	종료됨	1	2	종료됨	1	2	종료됨	1	2	종료됨
5	4	4	2	종료됨	1	4	4	2	종료됨	1	4	4	2	종료됨	1	4	4	2	종료됨	1	4	4	2
7	3	6	3	6	2	종료됨	1	3	6	3	6	2	종료됨	1	3	6	3	6	2	종료됨	1	3	6
11	10	5	5	5	10	10	10	5	2	종료됨	1	10	5	5	5	10	10	10	5	2	종료됨	1	10
13	12	3	6	4	12	12	4	3	6	12	2	종료됨	1	12	3	6	4	12	12	4	3	6	12
17	8	16	4	16	16	16	8	8	16	16	16	4	16	8	2	종료됨	1	8	16	4	16	16	16
19	18	18	9	9	9	3	6	9	18	3	6	18	18	18	9	9	2	종료됨	1	18	18	9	9
23	11	11	11	22	11	22	11	11	22	22	11	11	22	22	11	22	11	22	22	22	2	종료됨	1
29	28	28	14	14	14	7	28	14	28	28	4	14	28	28	7	4	28	28	7	28	14	7	7
31	5	30	5	3	6	15	5	15	15	30	30	30	15	10	5	30	15	15	15	30	30	10	30
37	36	18	18	36	4	9	12	9	3	6	9	36	12	36	9	36	36	36	36	18	36	12	36
41	20	8	10	20	40	40	20	4	5	40	40	40	8	40	5	40	5	40	20	20	40	10	40
43	14	42	7	42	3	6	14	21	21	7	42	21	21	21	7	21	42	42	42	7	14	21	21
47	23	23	23	46	23	23	23	23	46	46	23	46	23	46	23	23	23	46	46	23	46	46	23
53	52	52	26	52	26	26	52	26	13	26	52	13	52	13	13	26	52	52	52	52	52	4	13
59	58	29	29	29	58	29	58	29	58	58	29	58	58	29	29	29	58	29	29	29	29	58	58
61	60	10	30	30	60	60	20	5	60	4	15	3	6	15	15	60	60	30	5	12	15	20	20
67	66	22	33	22	33	66	22	11	33	66	66	66	11	11	33	33	66	33	66	33	11	33	11
71	35	35	35	5	35	70	35	35	35	70	35	70	10	35	35	10	35	35	7	70	70	14	35
73	9	12	9	72	36	24	3	6	8	72	36	72	72	72	9	24	18	36	72	24	8	36	12
79	39	78	39	39	78	78	13	39	13	39	26	39	26	26	39	26	13	39	39	13	13	3	6
83	82	41	41	82	82	41	82	41	41	41	41	82	82	82	41	41	82	82	82	41	82	41	82
89	11	88	11	44	88	88	11	44	44	22	8	88	88	88	11	44	44	88	44	44	22	88	88
97	48	48	24	96	12	96	16	24	96	48	16	96	96	96	12	96	16	32	32	96	4	96	24
101	100	100	50	25	10	100	100	50	4	100	100	50	10	100	25	10	100	25	50	50	50	50	25
103	51	34	51	102	102	51	17	17	34	102	102	17	17	51	51	51	51	51	102	102	34	17	34
107	106	53	53	106	106	106	106	53	53	53	53	53	53	106	53	106	106	53	106	106	106	53	106
109	36	27	18	27	108	27	12	27	108	108	54	108	108	27	9	36	108	36	54	27	27	36	108
113	28	112	14	112	112	14	28	56	112	56	112	56	28	4	7	112	8	112	112	112	56	112	112
127	7	126	7	42	126	126	7	63	42	63	126	63	126	63	7	63	63	3	6	63	9	126	18
131	130	65	65	65	130	65	130	65	130	65	65	65	130	65	65	130	26	26	65	65	130	130	26
137	68	136	34	136	136	68	68	68	8	68	136	136	34	34	17	68	34	68	136	136	34	136	136
139	138	138	69	69	23	69	46	69	46	69	138	69	46	138	69	138	138	138	69	138	138	46	69

최대 24개의 기준에서 주어진 기간의 소수점 표(최대 24개)

특유의 프라임으로 볼드한 느낌을 준다.

밑의 마침표 길이	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
1	(iii)	2	3	2	5	2, 3	7	2	3	2, 5	11	2, 3	13	2, 7	3, 5	2	17	2, 3	19	2, 5	3, 7	2, 11	23
2	3	(iii)	5	3	7	(iii)	3	5	11	3	13	7	3, 5	(iii)	17	3	19	5	3, 7	11	23	3	5
3	7	13	7	31	43	19	73	7, 13	37	7, 19	157	61	211	241	7, 13	307	7	127	421	463	13	7, 79	601
4	5	5	17	13	37	5	5, 13	41	101	61	5, 29	5, 17	197	113	257	5, 29	5, 13	181	401	13, 17	5, 97	5, 53	577
5	31	11	11, 31	11, 71	311	2801	31, 151	11, 61	41, 271	3221	22621	30941	11, 3761	11, 4931	11, 31, 41	88741	41, 2711	151, 911	11, 61, 251	40841	245411	292561	346201
6	(iii)	7	13	7	31	43	19	73	7, 13	37	7, 19	157	61	211	241	7, 13	307	7	127	421	463	13	7, 79
7	127	1093	43, 127	19531	55987	29, 4733	127, 337	547, 1093	239, 4649	43, 45319	659, 4943	5229043	8108731	1743463	29, 43, 113, 127	25646167	449, 80207	701, 70841	29, 71, 32719	43, 631, 3319	16968421	29, 5336717	29, 239, 28771
8	17	41	257	313	1297	1201	17, 241	17, 193	73, 137	7321	89, 233	14281	41, 937	17, 1489	65537	41761	113, 929	17, 3833	160001	97241	73, 3209	139921	331777
9	73	757	19, 73	19, 829	19, 2467	37, 1063	262657	19, 37, 757	333667	1772893	37, 80749	1609669	397, 18973	541, 21061	19, 37, 73, 109	19, 1270657	991, 34327	523, 29989	64008001	85775383	127, 297613	19, 7792003	19, 2017, 4987
10	11	61	41	521	11, 101	11, 191	11, 331	1181	9091	13421	19141	11, 2411	71, 101	31, 1531	61681	11, 71, 101	11, 9041	11, 2251	152381	185641	224071	31, 41, 211	11, 5791
11	23, 89	23, 3851	23, 89, 683	12207031	23, 3154757	1123, 293459	23, 89, 599479	23, 67, 661, 3851	21649, 513239	15797, 1806113	23, 266981089	23, 419, 859, 18041	67, 4027, 1154539	67, 463, 2333, 8537	23, 89, 397, 683, 2113	2141993519227	23, 199, 16127, 51217	104281, 62060021	10778947368421	17513875027111	67, 353, 1176469537	3937230404603	67, 7349, 134367047
12	13	73	241	601	13, 97	13, 181	37, 109	6481	9901	13, 1117	20593	28393	37, 1033	13, 3877	97, 673	83233	229, 457	13, 769	13, 12277	61, 3181	157, 1489	37, 7549	13, 73, 349
13	8191	797161	2731, 8191	305175781	3433, 760891	16148168401	79, 8191, 121369	398581, 797161	53, 79, 265371653	1093, 3158528101	477517, 20369233	53, 264031, 1803647	157, 29914249171	53, 157483, 16655159	53, 157, 1613, 2731, 8191	212057, 2919196853	79, 521, 29759719289	599, 29251, 133338869	3121, 142559, 9690539	79, 189437, 516094151	79, 2003, 85107437663	47691619, 480393499	53, 6553, 15913, 6895253
14	43	547	29, 113	29, 449	29, 197	113, 911	43, 5419	29, 16493	909091	1623931	211, 13063	29, 22079	7027567	10678711	15790321	22796593	32222107	197, 226871	827, 10529	81867661	29, 43, 86969	71, 673, 2969	183458857
15	151	4561	151, 331	181, 1741	1171, 1201	31, 159871	631, 23311	31, 271, 4561	31, 2906161	195019441	61, 661, 9781	4651, 161971	31, 2851, 15511	61, 39225301	61, 151, 331, 1321	6566760001	31, 601, 558721	31, 211, 2460181	31, 3001, 261451	211, 9391, 18181	61, 858794191	74912328481	241, 17881, 24481
16	257	17, 193	65537	17, 11489	17, 98801	17, 169553	97, 257, 673	21523361	17, 5882353	17, 6304673	17, 97, 260753	407865361	17, 5393, 16097	7121, 179953	641, 6700417	18913, 184417	97, 113607841	15073, 563377	17, 1505882353	62897, 300673	17, 3227992561	17, 3697, 623009	17, 2801, 2311681
17	131071	1871, 34511	43691, 131071	409, 466344409	239, 409, 1123, 30839	14009, 2767631689	103, 2143, 11119, 131071	103, 307, 1021, 1871, 34511	2071723, 5363222357	50544702849929377	2693651, 74876782031	103, 443, 15798461357509	103, 22771730193675277	1045002649, 6734509609	137, 953, 26317, 43691, 131071	10949, 1749233, 2699538733	7563707819165039903	3044803, 99995282631947	689852631578947368421	1502097124754084594737	239, 74729519, 176634767651	103, 62246266355102810647	307, 120574031, 341563234253
18	19	19, 37	37, 109	5167	46441	117307	87211	530713	19, 52579	590077	1657, 1801	19, 271, 937	19, 132049	19, 739, 811	433, 38737	1423, 5653	73, 465841	199, 236377	307, 69481	19, 37, 199, 613	19, 5966803	163, 271, 1117	127, 199, 7561
19	524287	1597, 363889	174763, 524287	191, 6271, 3981071	191, 638073026189	419, 4534166740403	32377, 524287, 1212847	1597, 2851, 101917, 363889	1111111111111111111	6115909044841454629	29043636306420266077	12865927, 9468940004449	459715689149916492091	4272113, 370649274902657	229, 457, 174763, 524287, 525313	229, 1103, 202607147, 291973723	6841, 6089884909802812423	109912203092239643840221	75368484119, 192696104561	12061389013, 54921106624003	45943, 341203, 97404596002423	2129, 63877469, 24939218613613	7282588256957615350925401
20	41	1181	61681	41, 9161	241, 6781	281, 4021	41, 61, 1321	42521761	3541, 27961	212601841	85403261	421, 601, 641	1061, 1383881	19421, 131381	4278255361	21881, 63541	15101, 145501	16936647121	41, 2801, 222361	41, 920421641	181, 401, 150901	61, 941, 272341	61, 1801385941
21	337	368089	337, 5419	379, 519499	1822428931	11898664849	92737, 649657	43, 2269, 368089	43, 1933, 10838689	1723, 8527, 27763	8177824843189	43, 337, 547, 2714377	43, 547, 2239000891	43, 2817034275427	337, 1429, 5419, 14449	43, 13567, 940143709	156107192084257	30640261, 68443621	460951, 8442733531	4789, 6427, 227633407	12271836836138419	43, 170689, 408030421	43, 10426753, 78066619
22	683	67, 661	397, 2113	23, 67, 5281	51828151	23, 10746341	67, 683, 20857	5501, 570461	23, 4093, 8779	23, 89, 199, 58367	57154490053	128011456717	23, 11737870057	23, 23504771357	353, 2931542417	23, 947, 87415373	536801, 6301307	23, 253239693257	23, 424016563147	23, 6073, 10362529	89, 285451051007	39700406579747	60867245726761
23	47, 178481	47, 1001523179	47, 178481, 2796203	8971, 332207361361	47, 139, 3221, 7505944891	47, 3083, 31479823396757	47, 178481, 10052678938039	47, 1001523179, 23535794707	11111111111111111111111	829, 28878847, 3740221981231	47, 39891250417, 321218438243	1381, 2519545342349331183143	47, 461, 2347, 10627, 2249861, 14525237	829, 31741, 3046462151831565769	47, 277, 1013, 1657, 30269, 178481, 2796203	47, 26552618219228090162977481	47, 599, 7468009, 20801237997245359	277, 2347, 16497763013, 1335495402823	691, 1381, 46266279097921483078651	47, 19597, 139870566115103282847737	4463, 1323064018651, 60575166785239	461, 1289, 831603031789, 1920647391913	47, 124799, 304751, 58769065453824529
24	241	6481	97, 673	390001	1678321	73, 193, 409	433, 38737	97, 577, 769	99990001	10657, 20113	193, 2227777	815702161	1475750641	2562840001	193, 22253377	73, 1321, 72337	11019855601	4297, 3952393	31177, 821113	73, 518118697	191353, 286777	937, 83575993	97, 1134793633

소수점 고유 소수점

현재 50개 이상의 고유 소수 또는 개연성이 있는 소수들이 알려져 있다.그러나 10 이하의¹⁰⁰ 고유한 소수만이 23개에 불과하다.다음 표에는 10 미만의¹⁰⁰ 23개의 고유 소수(순서 A040017(순서) 및 OEIS의 기간별 A007615(순서 길이에 따라 정렬) 및 그 기간(순서 A051627(순서:순서) 및 OEIS의 A007498(순서)이 나열되어 있다.

기간 길이	프라임
1	3
2	11
3	37
4	101
10	9,091
12	9,901
9	333,667
14	909,091
24	99,990,001
36	999,999,000,001
48	9,999,999,900,000,001
38	909,090,909,090,909,091
19	1,111,111,111,111,111,111
23	11,111,111,111,111,111,111,111
39	900,900,900,900,990,990,990,991
62	909,090,909,090,909,090,909,090,909,091
120	100,009,999,999,899,989,999,000,000,010,001
150	10,000,099,999,999,989,999,899,999,000,000,000,100,001
106	9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93	900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134	909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294	142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196	999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

기간 길이 294의 전성기는 7의 역수(0.14285714285728572857...)와^{[clarification needed]} 유사하다.

식탁 바로 뒤에 있는 24개의 고유 프라임은 128자리의 숫자와 320개의 마침표를 가지고 있다.(90₃₂₃₂)₂ + 1로 쓸 수 있으며, 여기서 첨자 번호 n은 첨자 앞에 숫자 또는 숫자 그룹의 연속적인 복사본이 n개임을 나타낸다.

드물기는 하지만, 리퓨즈 프리임의 발생과 개연성 프리임의 발생을 근거로 하여, 무한히 많은 독특한 프리임이 있다고 강하게 추측하고 있다.(모든 단위 프라임은 독특하다.)

2010년^[update] 현재, 재단위(10²⁷⁰³⁴³ – 1)/9는 알려진 가장 큰 개연성 소수점이다.^[2]

1996년에 가장 큰 것으로 입증된 고유 소수점은 (10¹¹³² + 1)/10001 또는 위의 표기법을 사용한 (9990000)₁₄₁ + 1이었다. 1128자리의 숫자를 가지고 있다.그 이후 기록이 여러 차례 개선되었다.2021년^[update] 현재 가장 큰 것으로 입증된 고유 프라임은 $\Phi _{11867}(-100)$ $\Phi _{11867}(-100)$ ( $\Phi _{11867}(-100)$ - $\Phi _{11867}(-100)$ ) $\Phi _{11867}(-100)$ 이며 $\Phi _{11867}(-100)$ 23732자리수를 가지고 있다.^[3]

바이너리 고유 소수

이진수(기본값 2)의 첫 번째 고유 소수점은 다음과 같다.

3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 43, 73, 127, 151, 241, 257, 331, 337, 683, ...(시퀀스 A144755(모양) 및 A161509(주기 길이순으로 정렬됨) (OEIS의 경우)

기간 길이는 다음과 같다.

2, 4, 4, 3, 10, 12, 8, 18, 5, 20, 9, 7, 15, 24, 16, 30, 21, 22, ...(시퀀스 A247071(해당 premimes로 주문됨) 및 A161508(변형됨)은 OEIS에서 제공된다.

페르마 프라임(기간 길이는 2의 힘), 메르센 프라임(기간 길이는 프라임), 바그스타프 프라임(기간 길이는 홀수 프라임의 2배) 등이 그것이다.

또한, 만약 n이 1 또는 6과 같지 않은 자연수라면, 지그몬디 정리 때문에 적어도 하나의 prime이 base 2에서 n 기간을 갖는 것보다 낫다.만약 n4(모드 8)와 엔>-20%, 동일한 것이다 게다가, 그 다음에 두 최고급 제품 베이스 2에,(베이스 2그러므로 n이 아닌 것은 특별한 기간)은Aurifeuillean 인수 분해 예를 들어, 113세()Φ 28L(2){\displaystyle =\Phi_{28L}(2)})과 29일()Φ 28M(2){\displaystyle =\Phi_{28M}(2)때문에})pe이 시대 n을 가지고 있다.리od 28 in base 2, 37 ( $=\Phi _{36L}(2)$ ) and 109 ( $=\Phi _{36M}(2)$ ) both have period 36 in base 2, and that 397 ( $=\Phi _{44L}(2)$ ) and 2113 ( $=\Phi _{44M}(2)$ ) both have period 44 in base 2,

위와 같이 prime p는 다음과 같은 자연수 c가 존재하는 경우에만 base 2에서 prime n의 고유한 prime이다.

{\frac {\Phi _{n}(2)}{\gcd(\Phi _{n},n)}}=p^{c}.

Empirically, it appears that most cases correspond to the denominator $\gcd(\Phi _{n}(2),n)$ being 1, so we have a prime (or prime power) of form $\Phi _{n}(2)$ . However, for n equal to 18, 20, 21, 54, 147, ... (sequence A333973 in the OEIS), we get a binary unique $\Phi _{n}(2)$ ( $\Phi _{n}(2)$ ) ${\displaystyle \Phi _{n}(2)$ 에 작은 구분자(n과 공유)가 있더라도 $\Phi _{n}(2)$ prime.

실제로 c > 1(즉 $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ ( 2 $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ ) $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ / $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ ( $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ ( 2 $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ ) $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ , n ) ${\displaystyle \Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ 은 $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ p의 참된 힘이며, 알려진 모든 고유 p는 c = 1이다.모든 고유 소수에는 c = 1(즉, 모든 베이스-2 고유 소수들은 위페리히 소수들이 아니다)이 있다고 추측된다.

2019년 9월 현재 가장 큰 것으로 알려진 베이스 2 유니크 프라임이 2-1이며^82589933, 또한 가장 큰 것으로 알려진 프라임이기도 하다.메르센 프라임을 제외하고 가장 큰 것으로 알려진 예상 베이스 2 고유 프라임은 2 ${\frac {2^{13372531}+1}{3}}$ + ${\frac {2^{13372531}+1}{3}}$ ${\frac {2^{13372531}+1}{3}}$ ${\$ { $2^{13372531}+1}{3$ } $}}}$ ,^[4] and the largest proven base 2 unique prime is ${\frac {2^{83339}+1}{3}}$ . Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is ${\frac {2^{4101572}+1}{17}}$ .

베이스 10과 유사하지만, 베이스 2에서는 모든 메르센 프리임이 독특하고, 메르센 프리메스가 무한히 많은 것으로 추측되기 때문에 베이스 2 고유의 프리메임이 무한히 많은 것으로 추측된다.^{[citation needed]}

하지만, 그 때문에 만일 복합 숫자Φ n로(2)gcd(Φ n(2), nx{\displaystyle{\frac{\Phi_{n}(2)}{\gcd(\Phi_{n}(2),n)}}}, 2를 두는.overpseudoprime은[설명 nee 쓸 수 있는 모든 다른 이상한 소수, 한 overpseudoprime 2기지로 나누overpseudoprimes 2기지로 중 누구도 나누었다.덩치]

베이스⁶⁴ 2 아래 2에는 52개의 고유한 프리타임이 있으며, 이 프리타임은 다음과 같다.

기간 길이	소수점(십진수)	Prime(이진수)
2	3	11
4	5	101
3	7	111
10	11	1011
12	13	1101
8	17	1 0001
18	19	1 0011
5	31	1 1111
20	41	10 1001
14	43	10 1011
9	73	100 1001
7	127	111 1111
15	151	1001 0111
24	241	1111 0001
16	257	1 0000 0001
30	331	1 0100 1011
21	337	1 0101 0001
22	683	10 1010 1011
26	2,731	1010 1010 1011
42	5,419	1 0101 0010 1011
13	8,191	1 1111 1111 1111
34	43,691	1010 1010 1010 1011
40	61,681	1111 0000 1111 0001
32	65,537	1 0000 0000 0000 0001
54	87,211	1 0101 0100 1010 1011
17	131,071	1 1111 1111 1111 1111
38	174,763	10 1010 1010 1010 1011
27	262,657	100 0000 0010 0000 0001
19	524,287	111 1111 1111 1111 1111
33	599,479	1001 0010 0101 1011 0111
46	2,796,203	10 1010 1010 1010 1010 1011
56	15,790,321	1111 0000 1111 0000 1111 0001
90	18,837,001	1 0001 1111 0110 1110 0000 1001
78	22,366,891	1 0101 0101 0100 1010 1010 1011
62	715,827,883	10 1010 1010 1010 1010 1010 1010 1011
31	2,147,483,647	111 1111 1111 1111 1111 1111 1111 1111
80	4,278,255,361	1111 1111 0000 0000 1111 1111 0000 0001
120	4,562,284,561	1 0000 1111 1110 1110 1111 0000 0001 0001
126	77,158,673,929	1 0001 1111 0111 0000 0011 1110 1110 0000 1001
150	1,133,836,730,401	1 0000 0111 1111 1101 1110 1111 1000 0000 0010 0001
86	2,932,031,007,403	10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
98	4,363,953,127,297	11 1111 1000 0000 1111 1110 0000 0011 1111 1000 0001
49	4,432,676,798,593	100 0000 1000 0001 0000 0010 0000 0100 0000 1000 0001
69	10,052,678,938,039	1001 0010 0100 1001 0010 0101 1011 0110 1101 1011 0111
65	145,295,143,558,111	1000 0100 0010 0101 0010 1001 0110 1011 0101 1011 1101 1111
174	96,076,791,871,613,611	1 0101 0101 0101 0101 0101 0101 0100 1010 1010 1010 1010 1010 1010 1011
77	581,283,643,249,112,959	1000 0001 0001 0010 0010 0110 0100 1100 1101 1001 1011 1011 0111 0111 1111
93	658,812,288,653,553,079	1001 0010 0100 1001 0010 0100 1001 0011 0110 1101 1011 0110 1101 1011 0111
122	768,614,336,404,564,651	1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
61	2,305,843,009,213,693,951	1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111
85	9,520,972,806,333,758,431	1000 0100 0010 0001 0100 1010 0101 0010 1011 0101 1010 1101 0111 1011 1101 1111
192	18,446,744,069,414,584,321	1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0001

표 뒤에 다음 10개의 바이너리 고유 프라임은 170, 234, 158, 165, 147, 129, 184, 89, 208, 312의 기간을 가진다.이외에도 비트(이진수)는 65, 73, 78, 81, 82, 84, 88, 89, 96, 97이다.

바이 유니크 프라임

바이 유니크 프리타임은 한 기간의 길이를 서로 공유하는 프리타임의 쌍이지만 다른 프리타임은 없다.예를 들어, 2진수에서 최소 1개의 프라이밍이 10000 미만인 바이 유니크 프리타임은 다음과 같다.

전성기의 p	p와 같은 기간을 가진 유일한 다른 프라임	마침표 길이
23	89	11
29	113	28
37	109	36
47	178481	23
59	3033169	58
61	1321	60
67	20857	66
71	122921	35
79	121369	39
83	8831418697	82
89	23	11
97	673	48
107	28059810762433	106
109	37	36
113	29	28
139	168749965921	138
167	57912614113275649087721	83
193	22253377	96
223	616318177	37
251	4051	50
263	10350794431055162386718619237468234569	131
281	86171	70
283	165768537521	94
353	2931542417	88
397	2113	44
433	38737	72
463	4982397651178256151338302204762057	231
571	160465489	114
577	487824887233	144
601	1801	25
607	1512768222413735255864403005264105839324374778520631853993	303
631	23311	45
641	6700417	64
643	84115747449047881488635567801	214
673	97	48
727	1786393878363164227858270210279	121
751	2139731020464054092520609592459940706818275139793055476751	375
769	442499826945303593556473164314770689	384
919	75582488424179347083438319	153
1039	19709014643115560219397264671577125505264032974428376489237001990435774189483906244488746953221813209	519
1291	83861817871925183739792206470703862766563053456867813459969184678546547694793573468589875745315081	1290
1321	61	60
1327	2365454398418399772605086209214363458552839866247069233	221
1429	14449	84
1471	252359902034571016856214298851708529738525821631	245
1543	4965395030068548134274243124972075225434447114375481299036593442726326832727934403424309955102162841656341524725641213163998408700663382552888660520657	771
1697	99335205800663868215396640964567095667094665346141013294320587365443384719802857319737050495099341955640963272958071602273	848
1753	1795918038741070627	146
1777	25781083	74
1801	601	25
2113	397	44
2281	3011347479614249131	190
2801	1114513219367157067542813609361306957257890531134775327875067038594481393220804051366788787128409731513666376851495151281817670381468528387601	1400
2971	48912491	110
3011	631215008947706187342830494125660733360092019659681922883823392015121754384870744044074337887482936870852519582960673945561810148710850934449712549090934572292098088972061029650939105592263256293676274598529593937386833315889748213948490958132757432166701901197169972066727635929332437543971934775961	3010
3259	960843850986532976532466235773483492840618819232206145010143480044702708779967241439519037158800917230289	1086
3361	88959882481	168
3967	3296810823331827444014404831943558588631803435050404237042485765714486337505843011741487225539321479275976317423474114853376321380782906502106758766783934866952124117240484839332668914566806988602931402117416523955329423560856334826333176954575294550104263404414368761262079586842542586869780254842277261781328657636993064897732127711363870426953852536828242291991249685206783121190349820804553	1983
4051	251	50
4129	33770734168253651800370989375796994825389296318018601048482005531172856260013942500368975908606689	688
4177	9857737155463	87
4523	106788290443848295284382097033	266
4561	51049903050598156013062477654241640657829025002976204451060261008689478158715729745160924860467530309657376827104233308157772350164622158651187694109112727796663977157921	2280
4871	82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337	487
5153	54410972897	112
5281	860573414369008969457638101533827364704684164286188824383996871471626764468219429432850649798234488791036977295952185305281368586922360240161830570497885830241813162641447900350702795124321	2640
5347	242099935645987	198
5431	9496792988973395279834809661569251320544014305629535339041176003993804676485199470146061766714674245857484692035146769896735263094609746643655454990307740324793870789211183639302657602006206727998357155351588374904926882678074633252877772442134938712719216422825919305415646969838444956229559805502059906228691551284152908049964563516405524127028294884716508273187514617905113012642316992618568629664046915514871575875088457784721	2715
5881	23618256244840618857212522155851714598259422753496906641681177748710460515038403366198473773770441	1470
6043	4475130366518102084427698737	318
6659	67348091890626757137914773048080151982788009808953349522971703676209072447919253736713943719839321930921001920443146832964494535806286153336758808764827052922284527987735369082658226155752758837734585788583920437031679967832798697874531506375848295306002069974292647012532975382657869395896549536084026643086849168707638035984652951756449232381922841508279043449553683642165895518852273616853752192626547082225232322662203349617421624450106130361133033050996977517456780759336980017504035553443204836636663583950658161718264375035034007950563418777684540446570742277682688819816930529249402141227674467861784042184664703273065772145626307008333021727910295689307897812592176340797189662547498298629041419687123412984210580325376481846316396566413701109848755348878790562296275295266190788880122151835567771665815565611863261470167857288685014242115505182159685753576612239477286620238583071292970734389580521730540789853959607322402465845627773409459421340250476125665859926003121138412497735360569	6658
6719	215006106257113223254503015023149432126193150293791416185445173578281597218315377296589584591228602041183907532584815068471747291177386898925622477208530115714962355294842135137890474394949339249259335407710018584480055157825387089416912233252714054247018216597994795059161567922302450277281351135838393171424038832688432240078361264161523904355539085927738753968157018550258476163852090826756157915705283413226000816151712543838581066281600650278690534719371112997393190721068136840596790525950480851370510277560248341182341805553054000587378384785994695875587394905226703149605689830257768229987714770949192483302583569799141079867597051190134078718011730508482542567284418838119000563443985593691221203060137047648713095502877775290062907208508727269017130916691676838817452529964938878349395785642430571852241837461604136374448443730175081889502056290512497717177577492736555784081731998565765598518104822516520340301701034123926767472784665776779480628821628279687651736198541330802238405154786248043073	3359
7487	26828803997912886929710867041891989490486893845712448833	197
8929	197107422273014301919781414466039325387889623676342705850752210599969	496
8969	10508537584872980049787749414505440238543661684506416445249892188329191267897669657242625405655025902294996965713681247700894953567276596965114308183649957469931262029470372188492494505614207827774171575432114297123003373257035070542940532411186322417809411123684246738342720455933424175399671044286557638075591	1121
9547	1621441292160739312484402643488810210953460916758334047593952342310982348899125523375207637304333778211869062392988099059802528019593682234941755422758885656395068722385037980657466257618112582188770921312100125511337836412531718154395821529210922389443733616354268820219577863577759459082447218927273695668223251258943006743614909639761127161704816862626236353032622115795192245125083091261029053988053316433377173895018793740052548266015018756763150731725385456332232982433576015547722563978072554378000015707071821371450842910648052930276764535303424167478747579771592484270800978561959411183367133498969236434846108865206764889977963554070295936092795484663326925724277620386077381551473009733178990983013487085676185144378484902884955972104873606558173086267499547566081780818064857671567480196001693236835136811036110768546793929610732909274227296407079545788520811837495181586420117807667033593394473	9546

10000 이하의 홀수가 1228개지만 이 중 21개만 특이하고 76개는 바이너리 바이 유니크다.

바이너리 바이 유니크 프리타임의 전형적인 예는 다음과 같다.

4681722635107226562076706769762761701618979214283287506897638413131392369211681565151468151465172014681547140045852280320356433(자릿수)

그리고

527739642811233917558838216073534609312522896254707972010583175760467054896492872702786549764052643493511382273226052631979775533936351462037464331880467187717179256707148303247 (177 digits)

메르센 2-1의¹⁰⁶¹ 두 주요 요인이다.^[5] 그러므로, 그 기간은
그들의 길이는 1061이다.

2016년 10월 현재 가장 큰 것으로 알려진 바이너리 바이 유니크 프라임은 ${\frac {2^{5240707}-1}{75392810903}}$ ${\frac {2^{5240707}-1}{75392810903}}$ - 1 ${\frac {2^{5240707}-1}{75392810903}}$ ${\$ 이며 ${\frac {2^{5240707}-1}{75392810903}}$ 기간이 있다.
주식이 75392810903인 5240707주.

마찬가지로, 우리는 "트리 유니크 프리타임"을 기간을 갖는 프리타임의 3배로 정의할 수 있다.
다른 소수만이 공유하는 길이2진수에서 처음 몇 개의 삼위일체 프리타임은 다음과 같다.

prime p	p와 같은 기간을 가진 유일한 다른 두 프라임	마침표 길이
53	157, 1613	52
101	8101, 268501	100
103	2143, 11119	51
131	409891, 7623851	130
137	953, 26317	68
157	53, 1613	52
163	135433, 272010961	162
179	62020897, 18584774046020617	178
181	54001, 29247661	180
191	420778751, 30327152671	95
197	19707683773, 4981857697937	196
199	153649, 33057806959	99
211	664441, 1564921	210
229	457, 525313	76
233	1103, 2089	29
271	348031, 49971617830801	135
307	2857, 6529	102
317	381364611866507317969, 604462909806215075725313	316
359	1433, 1489459109360039866456940197095433721664951999121	179
367	55633, 37201708625305146303973352041	183
373	951088215727633, 4611545283086450689	372
419	3410623284654639440707, 1607792018780394024095514317003	418
421	146919792181, 1041815865690181	420
431	9719, 2099863	43
439	2298041, 9361973132609	73
443	4714692062809, 4507513575406446515845401458366741487526913	442
457	229, 525313	76
467	27961, 352369374013660139472574531568890678155040563007620742839120913	466
491	15162868758218274451, 50647282035796125885000330641	490

2진수에서 가장 작은 n-유니크 프라임은

3, 23, 53, 149, 269, 461, 619, 389, ...

2진수에서 홀수의 기간 길이는 다음과 같다: (OEIS에서 연속 A014664).

전성기의	마침표 길이	전성기의	마침표 길이	전성기의	마침표 길이	전성기의	마침표 길이	전성기의	마침표 길이	전성기의	마침표 길이	전성기의	마침표 길이
3	2	79	39	181	180	293	292	421	420	557	556	673	48
5	4	83	82	191	95	307	102	431	43	563	562	677	676
7	3	89	11	193	96	311	155	433	72	569	284	683	22
11	10	97	48	197	196	313	156	439	73	571	114	691	230
13	12	101	100	199	99	317	316	443	442	577	144	701	700
17	8	103	51	211	210	331	30	449	224	587	586	709	708
19	18	107	106	223	37	337	21	457	76	593	148	719	359
23	11	109	36	227	226	347	346	461	460	599	299	727	121
29	28	113	28	229	76	349	348	463	231	601	25	733	244
31	5	127	7	233	29	353	88	467	466	607	303	739	246
37	36	131	130	239	119	359	179	479	239	613	612	743	371
41	20	137	68	241	24	367	183	487	243	617	154	751	375
43	14	139	138	251	50	373	372	491	490	619	618	757	756
47	23	149	148	257	16	379	378	499	166	631	45	761	380
53	52	151	15	263	131	383	191	503	251	641	64	769	384
59	58	157	52	269	268	389	388	509	508	643	214	773	772
61	60	163	162	271	135	397	44	521	260	647	323	787	786
67	66	167	83	277	92	401	200	523	522	653	652	797	796
71	35	173	172	281	70	409	204	541	540	659	658	809	404
73	9	179	178	283	94	419	418	547	546	661	660	811	270

2진수에서 주어진 기간 길이의 프리타임은 다음과 같다: (OEIS에서 순차 A108974).

마침표 길이	전성기	마침표 길이	전성기	마침표 길이	전성기	마침표 길이	전성기
1	(iii)	26	2731	51	103, 2143, 11119	76	229, 457, 525313
2	3	27	262657	52	53, 157, 1613	77	581283643249112959
3	7	28	29, 113	53	6361, 69431, 20394401	78	22366891
4	5	29	233, 1103, 2089	54	87211	79	2687, 202029703, 1113491139767
5	31	30	331	55	881, 3191, 201961	80	4278255361
6	(iii)	31	2147483647	56	15790321	81	2593, 71119, 97685839
7	127	32	65537	57	32377, 1212847	82	83, 8831418697
8	17	33	599479	58	59, 3033169	83	167, 57912614113275649087721
9	73	34	43691	59	179951, 3203431780337	84	1429, 14449
10	11	35	71, 122921	60	61, 1321	85	9520972806333758431
11	23, 89	36	37, 109	61	2305843009213693951	86	2932031007403
12	13	37	223, 616318177	62	715827883	87	4177, 9857737155463
13	8191	38	174763	63	92737, 649657	88	353, 2931542417
14	43	39	79, 121369	64	641, 6700417	89	618970019642690137449562111
15	151	40	61681	65	145295143558111	90	18837001
16	257	41	13367, 164511353	66	67, 20857	91	911, 112901153, 23140471537
17	131071	42	5419	67	193707721, 761838257287	92	277, 1013, 1657, 30269
18	19	43	431, 9719, 2099863	68	137, 953, 26317	93	658812288653553079
19	524287	44	397, 2113	69	10052678938039	94	283, 165768537521
20	41	45	631, 23311	70	281, 86171	95	191, 420778751, 30327152671
21	337	46	2796203	71	228479, 48544121, 212885833	96	193, 22253377
22	683	47	2351, 4513, 13264529	72	433, 38737	97	11447, 13842607235828485645766393
23	47, 178481	48	97, 673	73	439, 2298041, 9361973132609	98	4363953127297
24	241	49	4432676798593	74	1777, 25781083	99	199, 153649, 33057806959
25	601, 1801	50	251, 4051	75	100801, 10567201	100	101, 8101, 268501

기간 길이

기간 길이 표 1 ~ 100(특이한 소수점은 굵음)

기간 길이	프라임즈	기간 길이	프라임즈	기간 길이	프라임즈	기간 길이	프라임즈	기간 길이	프라임즈
1	3	21	43, 1933, 10838689	41	83, 1231, 538987, 201763709900322803748657942361	61	733, 4637, 329401, 974293, 1360682471, 106007173861643, 7061709990156159479	81	163, 9397, 2462401, 676421558270641, 130654897808007778425046117
2	11	22	23, 4093, 8779	42	127, 2689, 459691	62	909090909090909090909090909091	82	2670502781396266997, 3404193829806058997303
3	37	23	11111111111111111111111	43	173, 1527791, 1963506722254397, 2140992015395526641	63	10837, 23311, 45613, 45121231, 1921436048294281	83	3367147378267, 9512538508624154373682136329, 346895716385857804544741137394505425384477
4	101	24	99990001	44	89, 1052788969, 1056689261	64	19841, 976193, 6187457, 834427406578561	84	226549, 4458192223320340849
5	41, 271	25	21401, 25601, 182521213001	45	238681, 4185502830133110721	65	162503518711, 5538396997364024056286510640780600481	85	262533041, 8119594779271, 4222100119405530170179331190291488789678081
6	7, 13	26	859, 1058313049	46	47, 139, 2531, 549797184491917	66	599144041, 183411838171	86	57009401, 2182600451, 7306116556571817748755241
7	239, 4649	27	757, 440334654777631	47	35121409, 316362908763458525001406154038726382279	67	493121, 79863595778924342083, 28213380943176667001263153660999177245677	87	4003, 72559, 310170251658029759045157793237339498342763245483
8	73, 137	28	29, 281, 121499449	48	9999999900000001	68	28559389, 1491383821, 2324557465671829	88	617, 16205834846012967584927082656402106953
9	333667	29	3191, 16763, 43037, 62003, 77843839397	49	505885997, 1976730144598190963568023014679333	69	277, 203864078068831, 1595352086329224644348978893	89	497867, 103733951, 104984505733, 5078554966026315671444089, 403513310222809053284932818475878953159
10	9091	30	211, 241, 2161	50	251, 5051, 78875943472201	70	4147571, 265212793249617641	90	29611, 3762091, 8985695684401
11	21649, 513239	31	2791, 6943319, 57336415063790604359	51	613, 210631, 52986961, 13168164561429877	71	241573142393627673576957439049, 45994811347886846310221728895223034301839	91	547, 14197, 17837, 4262077, 43442141653, 316877365766624209, 110742186470530054291318013
12	9901	32	353, 449, 641, 1409, 69857	52	521, 1900381976777332243781	72	3169, 98641, 3199044596370769	92	1289, 18371524594609, 4181003300071669867932658901
13	53, 79, 265371653	33	67, 1344628210313298373	53	107, 1659431, 1325815267337711173, 47198858799491425660200071	73	12171337159, 1855193842151350117, 49207341634646326934001739482502131487446637	93	900900900900900900900900900900990990990990990990990990990991
14	909091	34	103, 4013, 21993833369	54	70541929, 14175966169	74	7253, 422650073734453, 296557347313446299	94	6299, 4855067598095567, 297262705009139006771611927
15	31, 2906161	35	71, 123551, 102598800232111471	55	1321, 62921, 83251631, 1300635692678058358830121	75	151, 4201, 15763985553739191709164170940063151	95	191, 59281, 63841, 1289981231950849543985493631, 965194617121640791456070347951751
16	17, 5882353	36	999999000001	56	7841, 127522001020150503761	76	722817036322379041, 1369778187490592461	96	97, 206209, 66554101249, 75118313082913
17	2071723, 5363222357	37	2028119, 247629013, 2212394296770203368013	57	21319, 10749631, 3931123022305129377976519	77	5237, 42043, 29920507, 136614668576002329371496447555915740910181043	97	12004721, 92556179448994367391887834053878562534782033760810527051075248738484727059555245899601591
18	19, 52579	38	909090909090909091	58	59, 154083204930662557781201849	78	157, 6397, 216451, 388847808493	98	197, 5076141624365532994918781726395939035533
19	1111111111111111111	39	900900900900990990990991	59	2559647034361, 4340876285657460212144534289928559826755746751	79	317, 6163, 10271, 307627, 49172195536083790769, 3660574762725521461527140564875080461079917	99	199, 397, 34849, 362853724342990469324766235474268869786311886053883
20	3541, 27961	40	1676321, 5964848081	60	61, 4188901, 39526741	80	5070721, 19721061166646717498359681	100	60101, 7019801, 14103673319201, 1680588011350901

다양한 베이스의 유니크 프라임

밑의	고유 기간 길이(최대 4096개)
2	2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410,414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, 602, 607, 626, 690, 694, 712, 745, 795, 816, 889, 897, 909, 954, 990, 1106, 1192, 1224, 1230, 1279, 1384, 1386, 1402, 1464, 1512, 1554, 1562, 1600, 1670, 1683, 1727, 1781, 1834, 1904, 1990, 1992, 2008, 2037, 2203, 2281, 2298, 2353, 2406, 2456, 2499, 2536, 2838, 3006, 3074, 3217, 3415, 3418, 3481, 3766, 3817, 3927, ...
3	1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, 665, 672, 680, 707, 718, 747, 760, 782, 828, 875, 892, 974, 984, 987, 1037, 1058, 1070, 1073, 1080, 1091, 1154, 1248, 1367, 1426, 1440, 1462, 1524, 1598, 1623, 1627, 1863, 1985, 2132, 2188, 2196, 2340, 2460, 2508, 2626, 2640, 2739, 2856, 3092, 3158, 3262, 3315, 3326, 3482, 3638, 3982, 4018, 4036, ...
4	1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, ...
5	1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, 619, 620, 694, 798, 897, 929, 981, 992, 1064, 1134, 1230, 1670, 1807, 2094, 2162, 2369, 2516, 2649, 2988, 3407, 3888, ...
6	1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, 663, 764, 814, 825, 874, 1028, 1049, 1050, 1102, 1113, 1131, 1158, 1376, 1464, 1468, 1535, 1622, 1782, 1834, 1924, 2096, 2176, 2409, 2464, 2816, 3013, 3438, 3453, 3663, ...
7	3, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, 581, 650, 654, 730, 740, 759, 1026, 1047, 1065, 1460, 1660, 1699, 1959, 2067, 2260, 2380, 2665, 2890, 3238, 4020, ...
8	1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, 1002, 1389, 1746, 1794, 2040, 2418, 2790, 3894, 4077, ...
9	1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, 624, 720, 762, 1066, 1094, 1098, 1170, 1230, 1254, 1320, 1428, 1546, 2018, 2574, 2724, 2804, 2920, 3074, 3316, 3646, ...
10	1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, ...
11	2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552, 810, 855, 878, 907, 1114, 1131, 1220, 1272, 1431, 1470, 1568, 1614, 1688, 1696, 1907, 2029, 2136, 2288, 2535, 2577, ...
12	1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, 636, 675, 701, 793, 945, 1090, 1268, 1272, 1971, 2088, 2368, 2482, 2893, 2966, 3290, ...
13	2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570, 594, 737, 744, 855, 883, 991, 1021, 1193, 1222, 1615, 1628, 1838, 2032, 2146, 2302, 2530, 2830, 2958, 3030, 3528, 3671, 3885, ...
14	1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560, 584, 864, 1006, 1335, 1363, 1520, 1536, 1659, 1862, 1925, 2332, 2458, 2687, 3381, 3512, 3870, 3976, ...
15	3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552, 590, 1029, 1194, 1274, 1406, 1444, 1532, 1548, 1748, 1986, 2093, 2182, 2202, 2579, 2781, 3054, 3239, 3696, ...
16	2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502, 614, 634, 1120, 1266, 1530, 1898, 1912, 1972, 2548, 2770, 3738, 3850, ...
17	1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515, 589, 624, 666, 716, 724, 810, 815, 1232, 1490, 1934, 2106, 2391, 2732, 2904, 3462, 3912, 4053, ...
18	1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448, 665, 792, 801, 816, 975, 1165, 1176, 1267, 1466, 1513, 1882, 1920, 1998, 2194, 2272, 2643, 2800, 2946, 3434, 3504, 3813, 3866, 3957, ...
19	2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565, 602, 612, 892, 984, 1061, 1067, 1079, 1262, 1328, 2356, 3033, 3419, 3501, 3963, ...
20	1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574, 774, 980, 1052, 1139, 1338, 1418, 1474, 1487, 1594, 1902, 2326, 3112, 3520, 3808, 3830, ...
21	2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516, 671, 694, 788, 1788, 1943, 2343, 2742, 3031, 3135, ...
22	2, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454, 602, 687, 706, 744, 857, 1028, 1074, 1136, 1150, 1345, 1408, 1525, 1572, 1578, 1988, 2142, 2665, ...
23	2, 5, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343, 580, 662, 816, 820, 846, 1078, 1092, 1174, 1189, 1548, 1632, 2040, 2180, 2348, 2732, 3100, 3181, 4010, ...
24	1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430, 653, 661, 744, 834, 855, 870, 927, 1128, 1158, 1390, 1516, 1555, 1617, 1844, 2022, 2060, 2208, 2812, 3153, 3952, ...
25	2, 4, 6, 12, 14, 24, 28, 44, 46, 54, 58, 60, 118, 124, 144, 192, 210, 250, 268, 310, 496, 532, 1258, 1494, 1944, 2050, 2498, 2728, 3324, 3418, 3646, 3862, 4014, ...
26	1, 2, 4, 7, 9, 18, 20, 22, 24, 30, 43, 69, 132, 140, 186, 200, 210, 218, 267, 347, 454, 495, 554, 585, 645, 694, 980, 1028, 1060, 1098, 1432, 1714, 1828, 3513, 3786, ...
27	2, 3, 12, 21, 24, 36, 87, 93, 171, 249, 276, 360, 480, 621, 732, 780, 1716, 3843, ...
28	1, 2, 3, 5, 6, 8, 17, 21, 38, 81, 91, 96, 102, 132, 148, 156, 240, 258, 260, 276, 457, 464, 465, 500, 506, 535, 684, 746, 838, 930, 982, 1015, 1189, 1296, 1335, 1345, 1390, 1423, 2062, 2723, 2893, 3078, ...
29	4, 5, 6, 7, 8, 14, 30, 32, 39, 45, 50, 76, 116, 151, 222, 357, 402, 462, 570, 588, 636, 671, 695, 844, 1498, 1650, 1770, 3175, 3195, 3312, 3538, 3719, ...
30	1, 2, 5, 9, 11, 12, 21, 36, 51, 64, 91, 163, 174, 195, 230, 278, 318, 342, 346, 424, 530, 569, 578, 795, 984, 1094, 1167, 1335, 1564, 1605, 1658, 1789, 2159, 2204, 2225, 3366, 3458, 3615, ...
31	3, 7, 12, 17, 24, 30, 31, 33, 40, 176, 218, 308, 404, 420, 630, 693, 890, 915, 922, 1475, 2122, 2185, 2487, 2541, 2907, 3387, 4055, ...
32	1, 6, 30, 85, 110, 120, 320, 1050, 1065, 1385, 2490, 3080, 3920, ...
33	1, 2, 3, 10, 16, 25, 28, 30, 35, 36, 45, 56, 76, 87, 110, 134, 135, 197, 200, 220, 228, 314, 324, 330, 396, 498, 583, 624, 725, 806, 940, 1145, 1240, 1644, 1750, 2171, 2268, 2675, 2781, 2790, 2808, 3581, ...
34	3, 6, 8, 10, 13, 20, 24, 56, 87, 154, 164, 196, 282, 363, 428, 652, 744, 780, 860, 902, 952, 1178, 1493, 1540, 1643, 1904, 2184, 2277, 2468, 2943, ...
35	2, 4, 6, 8, 18, 21, 22, 26, 42, 128, 154, 158, 170, 180, 184, 192, 254, 313, 450, 624, 737, 762, 798, 874, 912, 1002, 1006, 1098, 1234, 1297, 1418, 1714, 1926, 2325, 2343, 2368, 2998, 3567, 4064, ...
36	2, 4, 12, 62, 72, 96, 180, 240, 382, 514, 688, 732, 734, 962, 1048, 1088, 1232, 1408, 2088, 2176, 2248, 2724, 3180, ...

n	기간 n과 함께 독특한 전성기가 있는 것을 기본으로 하다.
1	3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, 308, 312, 314, 318, 332, 338, 348, 350, 354, 360, 368, 374, 380, 384, 390, 398, 402, 410, 420, 422, 432, 434, 440, 444, 450, 458, 462, 464, 468, 480, 488, 492, 500, 504, 510, 522, 524, 542, 548, 558, 564, 570, 572, 578, 588, 594, 600, 602, 608, 614, 618, 620, 632, 642, 644, 648, 654, 660, 662, 674, 678, 684, 692, 702, 710, 720, 728, 734, 740, 744, 752, 758, 762, 770, 774, 788, 798, 810, 812, 822, 824, 828, 830, 840, 854, 858, 860, 864, 878, 882, 884, 888, 908, 912, 920, 930, 938, 942, 948, 954, 968, 972, 978, 984, 992, 998, 1010, 1014, 1020, 1022, 1032, 1034, 1040, 1050, 1052, 1062, 1064, 1070, 1088, 1092, 1094, 1098, 1104, 1110, 1118, 1124, 1130, 1152, 1154, 1164, 1172, 1182, 1188, 1194, 1202, 1214, 1218, 1224, 1230, 1232, 1238, 1250, 1260, 1278, 1280, 1284, 1290, 1292, 1298, 1302, 1304, 1308, 1320, 1322, 1328, 1362, 1368, 1374, 1382, 1400, 1410, 1424, 1428, 1430, 1434, 1440, 1448, 1452, 1454, 1460, 1472, 1482, 1484, 1488, 1490, 1494, 1500, 1512, 1524, 1532, 1544, 1550, 1554, 1560, 1568, 1572, 1580, 1584, 1598, 1602, 1608, 1610, 1614, 1620, 1622, 1628, 1638, 1658, 1664, 1668, 1670, 1694, 1698,1700, 1710, 1722, 1724, 1734, 1742, 1748, 1754, 1760, 1778, 1784, 1788, 1790, 1802, 1812, 1824, 1832, 1848, 1862, 1868, 1872, 1874, 1878, 1880, 1890, 1902, 1908, 1914, 1932, 1934, 1950, 1952, 1974, 1980, 1988, 1994, 1998, 2000, 2004, 2012, 2018, 2028, 2030, 2040, 2054, 2064, 2070, 2082, 2084, 2088, 2090, 2100, 2112, 2114, 2130, 2132, 2138, 2142, 2144, 2154, 2162, 2180, 2204, 2208, 2214, 2222, 2238, 2240, 2244, 2252, 2268, 2270, 2274, 2282, 2288, 2294, 2298, 2310, 2312, 2334, 2340, 2342, 2348, 2352, 2358, 2372, 2378, 2382, 2384, 2390, 2394, 2400, 2412, 2418, 2424, 2438, 2442, 2448, 2460, 2468, 2474, 2478, ...
2	2, 4, 5, 6, 9, 10, 11, 12, 13, 16, 18, 19, 21, 22, 23, 25, 27, 28, 30, 33, 36, 37, 39, 40, 42, 43, 45, 46, 47, 51, 52, 55, 57, 58, 60, 61, 66, 67, 70, 72, 73, 75, 78, 79, 81, 82, 85, 87, 88, 91, 93, 95, 96, 100, 102, 103, 105, 106, 108, 111, 112, 115, 117, 121, 123, 126, 130, 133, 135, 136, 138, 141, 145, 147, 148, 150, 151, 156, 157, 159, 162, 163, 165, 166, 171, 172, 175, 177, 178, 180, 183, 187, 190, 191, 192, 193, 196, 198, 201, 205, 207, 210, 211, 213, 217, 222, 223, 225, 226, 228, 231, 232, 235, 238, 240, 243, 247, 250, 253, 256, 261, 262, 267, 268, 270, 271, 273, 276, 277, 280, 282, 283, 291, 292, 295, 297, 301, 303, 306, 310, 312, 313, 315, 316, 319, 325, 327, 330, 331, 333, 336, 343, 345, 346, 348, 351, 352, 355, 357, 358, 361, 366, 367, 372, 375, 378, 381, 382, 383, 385, 387, 388, 393, 396, 397, 400, 403, 408, 411, 415, 418, 420, 421, 423, 427, 430, 432, 435, 438, 442, 445, 447, 448, 451, 453, 456, 457, 460, 462, 463, 465, 466, 471, 477, 478, 481, 486, 487, 490, 495, 498, 501, 502, 507, 508, 513, 520, 522, 523, 525, 535, 537, 540, 541, 543, 546, 547, 553, 555, 556, 561, 562, 565, 567, 568, 570, 576, 583, 585, 586, 591, 592, 595, 598, 600, 603, 606, 607, 612, 613, 616, 618, 621, 625, 627, 630, 631, 633, 639, 640, 642, 646, 651, 652, 655, 658, 660, 661, 663, 667, 672, 673, 676, 682, 687, 690, 691, 693, 697, 700, 703, 705, 708, 711, 715, 717, 718, 723, 726, 732, 733, 735, 738, 742, 745, 750, 751, 756, 757, 760, 763, 765, 767, 768, 771, 772, 775, 777, 786, 787, 793, 795, 796, 801, 807, 808, 810, 817, 820, 822, 823, 826, 828, 831, 837, 838, 841, 843, 847, 852, 855, 856, 858, 861, 862, 865, 871, 876, 877, 880, 882, 885, 886, 891, 895, 897, 903, 906, 907, 910, 913, 915, 918, 921, 925, 927, 928, 931, 933, 936, 940, 943, 946, 952, 955, 957, 963, 966, 970, 973, 975, 976, 981, 982, 990, 991, 996, 997, 1003, 1005, 1008, 1012, 1015, 1017, 1018, 1020, 1027, 1030, 1032, 1038, 1041, 1045, 1047, 1048, 1050, 1051, 1060, 1062, 1068, 1071, 1075, 1081, 1083, 1086, 1087, 1090, 1092, 1093, 1095, 1096, 1102, 1107, 1108, 1111, 1113, 1116, 1122, 1123, 1125, 1128, 1131, 1135, 1137, 1141, 1150, 1152, 1153, 1162, 1167, 1170, 1171, 1173, 1180, 1183, 1185, 1186, 1191, 1192, 1197, 1200, 1201, 1207, 1212, 1213, 1215, 1216, 1222, 1225, 1227, 1228, 1230, 1233, 1236, 1237, 1243, 1248, 1251, 1255, 1258, 1261, 1263, 1267, 1276, 1278, 1279, 1281, 1282, 1285, 1288, 1290, 1293, 1296, 1300, 1302, 1303, 1305, 1306, 1311, 1317, 1318, 1320,1321, 1323, 1326, 1327, 1335, 1345, 1347, 1353, 1360, 1365, 1366, 1372, 1375, 1380, 1381, 1383, 1387, 1395, 1398, 1401, 1407, 1408, 1411, 1417, 1422, 1423, 1426, 1428, 1431, 1432, 1435, 1437, 1438, 1446, 1447, 1450, 1452, 1453, 1458, 1465, 1467, 1470, 1471, 1477, 1480, 1482, 1485, 1486, 1488, 1491, 1492, 1498, 1501, 1503, 1510, 1513, 1515, 1521, 1522, 1527, 1530, 1531, 1535, 1537, 1542, 1543, 1545, 1548, 1551, 1552, 1555, 1558, 1566, 1570, 1573, 1575, 1578, 1582, 1587, 1591, 1593, 1596, 1600, 1603, 1606, 1608, 1612, 1615, 1617, 1618, 1620, 1621, 1626, 1635, 1636, 1641, 1645, 1647, 1653, 1656, 1657, 1662, 1663, 1666, 1668, 1675, 1677, 1683, 1687, 1692, 1695, 1696, 1698, 1705, 1708, 1711, 1713, 1717, 1720, 1722, 1723, 1725, 1731, 1732, 1740, 1743, 1746, 1752, 1753, 1755, 1758, 1761, 1765, 1771, 1773, 1776, 1782, 1783, 1786, 1788, 1791, 1795, 1800, 1807, 1810, 1813, 1815, 1821, 1822, 1827, 1830, 1831, 1837, 1843, 1846, 1851, 1855, 1857, 1860, 1863, 1866, 1867, 1870, 1872, 1873, 1876, 1878, 1881, 1887, 1888, 1893, 1900, 1905, 1906, 1911,1912, 1915, 1927, 1930, 1932, 1933, 1941, 1947, 1948, 1950, 1951, 1953, 1963, 1965, 1972, 1978, 1981, 1983, 1986, 1992, 1993, 1995, 1996, 1998, 2002, 2007, 2010, 2011, 2016, 2017, 2025, 2026, 2028, 2031, 2035, 2037, 2038, 2041, 2052, 2055, 2061, 2062, 2065, 2068, 2077, 2080, 2082, 2083, 2086, 2088, 2091, 2095, 2097, 2098, 2101, 2103, 2110, 2112, 2121, 2125, 2128, 2130, 2136, 2137, 2140, 2142, 2143, 2151, 2152, 2160, 2163, 2167, 2173, 2175, 2178, 2181, 2185, 2187, 2191, 2193, 2202, 2205, 2206, 2212, 2215, 2217, 2220, 2223, 2227, 2233, 2236, 2238, 2242, 2245, 2247, 2250, 2251, 2257, 2263, 2266, 2268, 2271, 2272, 2275, 2280, 2283, 2286, 2292, 2296, 2301, 2305, 2307, 2308, 2310, 2325, 2332, 2335, 2338, 2340, 2341, 2343, 2346, 2347, 2350, 2356, 2361, 2367, 2370, 2371, 2373, 2376, 2380, 2382, 2383, 2385, 2388, 2392, 2395, 2398, 2401, 2403, 2410, 2415, 2416, 2422, 2425, 2427, 2431, 2433, 2436, 2440, 2445, 2446, 2451, 2455, 2457, 2458, 2461, 2466, 2467, 2472, 2473, 2475, 2476, 2487, 2497, ...
3	2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 17, 19, 20, 21, 24, 27, 28, 31, 33, 34, 38, 40, 41, 43, 50, 52, 54, 57, 59, 62, 66, 69, 70, 71, 73, 75, 76, 77, 78, 80, 82, 85, 89, 90, 91, 97, 99, 101, 103, 105, 110, 111, 112, 115, 117, 119, 124, 127, 131, 136, 138, 141, 143, 145, 147, 148, 150, 153, 155, 157, 161, 162, 164, 166, 167, 168, 173, 175, 176, 187, 188, 189, 190, 192, 194, 199, 202, 203, 206, 209, 215, 218, 223, 231, 236, 241, 244, 245, 246, 259, 265, 266, 271, 272, 274, 278, 279, 280, 286, 287, 288, 290, 293, 309, 314, 316, 325, 329, 332, 336, 342, 344, 348, 351, 357, 358, 369, 370, 376, 378, 381, 383, 385, 388, 392, 395, 398, 402, 404, 405, 409, 414, 416, 421, 426, 427, 434, 435, 442, 447, 453, 455, 456, 460, 469, 472, 475, 476, 489, 493, 495, 496, 500, 511, 512, 514, 517, 518, 525, 526, 530, 531, 533, 537, 538, 540, 551, 553, 554, 556, 559, 560, 566, 567, 572, 574, 579, 580, 582, 584, 586, 589, 603, 605, 609, 612, 616, 621, 622, 624, 626, 628, 635, 637, 642, 644, 649, 658, 661, 668, 671, 673, 677, 682, 686, 696, 700, 701, 712, 715, 720, 726, 727, 728, 735, 736, 743, 747, 754, 755, 761, 762, 768, 773, 775, 778, 782, 785, 792, 798, 799, 801, 812, 818, 819, 825, 827, 832, 836, 838, 839, 846, 850, 855, 857, 859, 860, 864, 868, 875, 878, 883, 889, 890, 894, 895, 897, 899, 910, 911, 915, 916, 918, 920, 922, 925, 927, 937, 943, 946, 950, 959, 960, 964, 969, 974, 981, 987, 990, 992, 993, 994, 1000, 1001, 1002, 1006, 1007, 1011, 1016, 1020, 1021, 1022, 1027, 1029, 1039, 1041, 1051, 1057, 1058, 1060, 1063, 1065, 1067, 1070, 1072, 1074, 1077, 1078, 1091, 1092, 1093, 1097, 1098, 1105, 1107, 1116, 1118, 1125, 1126, 1130, 1133, 1142, 1144, 1146, 1148, 1149, 1163, 1165, 1174, 1181, 1182, 1191, 1193, 1195, 1196, 1198, 1200, 1202,1209, 1216, 1217, 1221, 1230, 1252, 1254, 1256, 1258, 1260, 1263, 1272, 1274, 1275, 1281, 1288, 1295, 1308, 1312, 1314, 1317, 1323, 1324, 1330, 1331, 1333, 1336, 1340, 1343, 1345, 1349, 1351, 1352, 1354, 1359, 1363, 1364, 1365, 1372, 1373, 1380, 1386, 1392, 1396, 1401, 1405, 1414, 1415, 1421, 1422, 1427, 1434, 1438, 1440, 1441, 1442, 1443, 1447, 1448, 1449, 1454, 1456, 1473, 1475, 1483, 1484, 1487, 1492, 1494, 1505, 1510, 1515, 1519, 1520, 1522, 1526, 1529, 1534, 1536, 1539, 1540, 1547, 1548, 1559, 1560, 1561, 1562, 1566, 1568, 1573, 1581, 1583, 1585, 1590, 1592, 1594, 1596, 1609, 1611, 1616, 1620, 1624, 1630, 1632, 1639, 1644, 1646, 1650, 1651, 1657, 1658, 1659, 1665, 1666, 1669, 1672, 1676, 1678, 1685, 1690, 1692, 1695, 1700, 1701, 1707, 1708, 1709, 1711, 1716, 1718, 1720, 1723, 1729, 1746, 1748, 1750, 1753, 1757, 1762, 1767, 1770, 1788, 1795, 1800, 1804, 1806, 1809, 1811, 1814, 1818, 1825, 1826, 1828, 1832, 1837, 1841, 1847, 1848, 1853, 1861, 1863, 1879, 1882, 1886, 1895, 1902, 1909, 1923, 1931, 1932, 1939, 1952, 1956, 1960, 1963,1970, 1973, 1974, 1977, 1982, 1986, 1993, 1994, 1996, 2000, 2010, 2015, 2016, 2022, 2029, 2033, 2043, 2049, 2054, 2056, 2058, 2064, 2065, 2068, 2073, 2075, 2079, 2080, 2085, 2086, 2091, 2105, 2107, 2110, 2112, 2117, 2129, 2136, 2138, 2140, 2142, 2145, 2157, 2159, 2162, 2163, 2168, 2170, 2171, 2176, 2183, 2191, 2194, 2197, 2201, 2203, 2204, 2205, 2227, 2233, 2236, 2238, 2241, 2247, 2255, 2259, 2260, 2262, 2264, 2273, 2275, 2276, 2280, 2290, 2296, 2301, 2303, 2309, 2313, 2318, 2324, 2339, 2345, 2353, 2359, 2360, 2364, 2372, 2373, 2380, 2385, 2387, 2390, 2397, 2402, 2409, 2411, 2422, 2429, 2430, 2435, 2437, 2442, 2446, 2450, 2451, 2455, 2456, 2463, 2464, 2465, 2470, 2471, 2476, 2478, 2483, 2484, 2491, 2493, 2495, 2497, 2498, ...
4	2, 3, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 24, 25, 26, 29, 35, 36, 39, 40, 45, 49, 51, 54, 56, 59, 61, 65, 66, 69, 71, 74, 79, 84, 85, 90, 94, 95, 101, 110, 116, 120, 121, 124, 126, 130, 131, 134, 139, 141, 145, 146, 150, 156, 159, 160, 165, 169, 170, 171, 175, 176, 180, 181, 184, 195, 199, 201, 204, 205, 206, 209, 210, 219, 221, 224, 230, 231,236, 240, 245, 250, 256, 260, 261, 264, 270, 271, 275, 279, 280, 284, 289, 299, 300, 306, 309, 314, 315, 321, 325, 326, 329, 335, 340, 345, 349, 350, 371, 375, 379, 384, 386, 391, 396, 399, 400, 405, 406, 409, 415, 420, 425, 430, 435, 436, 440, 441, 444, 445, 449, 451, 459, 461, 464, 466, 470, 471, 474, 490, 496, 519, 521, 529, 535, 536, 544, 545,556, 559, 569, 570, 571, 575, 576, 579, 581, 584, 594, 595, 609, 631, 634, 636, 639, 641, 644, 646, 649, 654, 661, 669, 674, 680, 685, 686, 689, 690, 695, 696, 699, 700, 704, 711, 714, 715, 716, 739, 740, 745, 750, 751, 760, 764, 779, 780, 781, 784, 791, 799, 815, 816, 819, 821, 826, 841, 855, 860, 861, 864, 869, 875, 881, 885, 890, 901, 906, 909,910, 920, 921, 925, 929, 930, 935, 936, 946, 949, 950, 951, 955, 959, 960, 966, 979, 981, 985, 986, 989, 991, 1001, 1004, 1010, 1011, 1025, 1029, 1031, 1036, 1039, 1051, 1054, 1055, 1060, 1066, 1069, 1070, 1080, 1081, 1091, 1094, 1095, 1096, 1099, 1106, 1111, 1124, 1125, 1129, 1140, 1144, 1146, 1150, 1151, 1155, 1156, 1161, 1171, 1174, 1176, 1179,1181, 1184, 1185, 1199, 1205, 1210, 1219, 1225, 1241, 1244, 1246, 1251, 1255, 1265, 1274, 1276, 1281, 1285, 1290, 1294, 1299, 1306, 1311, 1314, 1315, 1316, 1320, 1324, 1329, 1340, 1345, 1349, 1350, 1354, 1359, 1361, 1366, 1374, 1376, 1389, 1391, 1394, 1405, 1406, 1410, 1411, 1416, 1419, 1420, 1421, 1430, 1434, 1439, 1440, 1456, 1459, 1460, 1465, 1469, 1489, 1494, 1495, 1499, 1501, 1504, 1515, 1519, 1521, 1524, 1531, 1535, 1541, 1545, 1546, 1550, 1556, 1564, 1566, 1569, 1570, 1576, 1580, 1586, 1589, 1599, 1605, 1614, 1615, 1616, 1629, 1635, 1639, 1640, 1644, 1654, 1660, 1661, 1664, 1674, 1676, 1684, 1686, 1689, 1691, 1700, 1701, 1705, 1709, 1715, 1716, 1736, 1741, 1749, 1756, 1765, 1766, 1771, 1774, 1784, 1790, 1794, 1805, 1811, 1816, 1824, 1831, 1845, 1850, 1855, 1859, 1860, 1869, 1870, 1875, 1876, 1884, 1894, 1899, 1901, 1910, 1915, 1920, 1935, 1940, 1949, 1964, 1966, 1970, 1974, 1980, 1990, 2001, 2006, 2009, 2021, 2026, 2034, 2035, 2050, 2051, 2054, 2055, 2056, 2064, 2065, 2074, 2079, 2080, 2084, 2086, 2091, 2094, 2096, 2106, 2116,2120, 2125, 2126, 2135, 2136, 2139, 2145, 2151, 2154, 2159, 2161, 2165, 2171, 2174, 2175, 2199, 2201, 2210, 2224, 2239, 2255, 2260, 2261, 2266, 2269, 2271, 2281, 2285, 2286, 2289, 2294, 2304, 2310, 2314, 2315, 2320, 2321, 2326, 2330, 2331, 2336, 2341, 2349, 2351, 2354, 2360, 2375, 2379, 2380, 2381, 2389, 2391, 2404, 2405, 2406, 2420, 2425, 2430, 2454, 2456, 2459, 2460, 2464, 2470, 2481, 2496, ...
5	2, 3, 6, 7, 11, 12, 13, 17, 21, 22, 23, 24, 28, 29, 30, 40, 41, 43, 44, 46, 50, 56, 62, 63, 68, 73, 74, 77, 79, 83, 85, 94, 99, 110, 111, 117, 118, 120, 122, 127, 129, 134, 143, 145, 151, 154, 162, 164, 165, 166, 172, 175, 177, 186, 193, 198, 204, 206, 208, 222, 227, 239, 249, 254, 255, 260, 261, 263, 265, 274, 275, 276, 277, 285, 288, 292, 304, 308, 327, 336, 337, 340, 352, 359, 369, 373, 393, 397, 408, 414, 417, 418, 437, 439, 448, 457, 459, 474, 479, 486, 490, 492, 495, 496, 503, 505, 514, 519, 528, 530, 536, 538, 539, 540, 550, 556, 557, 561, 563, 567, 568, 572, 579, 594, 601, 604, 616, 617, 626, 637, 645, 650, 662, 679, 694, 699, 706, 711, 714, 728, 745, 750, 761, 765, 770, 772, 793, 804, 805, 824, 826, 837, 854, 860, 864, 868, 880, 886, 890, 891, 919, 921, 931, 936, 941, 942, 954, 967, 968, 974, 976, 979, 986, 991, 1001, 1007, 1009, 1044, 1045, 1047, 1053, 1066, 1078, 1079, 1085, 1102, 1108, 1111, 1129, 1133, 1135, 1152, 1154, 1157, 1173, 1178, 1183, 1187, 1201, 1209, 1228, 1239, 1249, 1251, 1253, 1266, 1276, 1282, 1284, 1297, 1327, 1333, 1343, 1348, 1352, 1370, 1377, 1382, 1396, 1398, 1416, 1419, 1426, 1429, 1452, 1462, 1463, 1475, 1480, 1481, 1493, 1503, 1506, 1520, 1525, 1536, 1553, 1557, 1562, 1563, 1579, 1596, 1641, 1646, 1657, 1662, 1663, 1668, 1672, 1679, 1683, 1684, 1694, 1695, 1696, 1704, 1746, 1748, 1749, 1755, 1760, 1762, 1767, 1770, 1788, 1826, 1836, 1838, 1843, 1859, 1881, 1883, 1891, 1914, 1921, 1927, 1944, 1947, 1960, 1975, 1977, 1979, 1981, 1987, 1990, 1991, 2021, 2032, 2034, 2037, 2041, 2052, 2059, 2069, 2076, 2080, 2086, 2090, 2097, 2098, 2111, 2118, 2123, 2131, 2134, 2164, 2167, 2175, 2177, 2218, 2228, 2233, 2243, 2245, 2252, 2261, 2272, 2274, 2289, 2295, 2299, 2300, 2305, 2318, 2321, 2328, 2332,2340, 2356, 2361, 2362, 2366, 2373, 2375, 2382, 2383, 2387, 2430, 2431, 2432, 2438, 2448, 2450, 2464, 2483, 2487, 2499, ...
6	3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 18, 20, 21, 22, 25, 28, 29, 32, 34, 35, 39, 41, 42, 44, 51, 53, 55, 58, 60, 63, 67, 70, 71, 72, 74, 76, 77, 78, 79, 81, 83, 86, 90, 91, 92, 98, 100, 102, 104, 106, 111, 112, 113, 116, 118, 120, 125, 128, 132, 137, 139, 142, 144, 146, 148, 149, 151, 154, 156, 158, 162, 163, 165, 167, 168, 169, 174, 176, 177,188, 189, 190, 191, 193, 195, 200, 203, 204, 207, 210, 216, 219, 224, 232, 237, 242, 245, 246, 247, 260, 266, 267, 272, 273, 275, 279, 280, 281, 287, 288, 289, 291, 294, 310, 315, 317, 326, 330, 333, 337, 343, 345, 349, 352, 358, 359, 370, 371, 377, 379, 382, 384, 386, 389, 393, 396, 399, 403, 405, 406, 410, 415, 417, 422, 427, 428, 435, 436, 443,448, 454, 456, 457, 461, 470, 473, 476, 477, 490, 494, 496, 497, 501, 512, 513, 515, 518, 519, 526, 527, 531, 532, 534, 538, 539, 541, 552, 554, 555, 557, 560, 561, 567, 568, 573, 575, 580, 581, 583, 585, 587, 590, 604, 606, 610, 613, 617, 622, 623, 625, 627, 629, 636, 638, 643, 645, 650, 659, 662, 669, 672, 674, 678, 683, 687, 697, 701, 702, 713,716, 721, 727, 728, 729, 736, 737, 744, 748, 755, 756, 762, 763, 769, 774, 776, 779, 783, 786, 793, 799, 800, 802, 813, 819, 820, 826, 828, 833, 837, 839, 840, 847, 851, 856, 858, 860, 861, 865, 869, 876, 879, 884, 890, 891, 895, 896, 898, 900, 911, 912, 916, 917, 919, 921, 923, 926, 928, 938, 944, 947, 951, 960, 961, 965, 970, 975, 982, 988, 991,993, 994, 995, 1001, 1002, 1003, 1007, 1008, 1012, 1017, 1021, 1022, 1023, 1028, 1030, 1040, 1042, 1052, 1058, 1059, 1061, 1064, 1066, 1068, 1071, 1073, 1075, 1078, 1079, 1092, 1093, 1094, 1098, 1099, 1106, 1108, 1117, 1119, 1126, 1127, 1131, 1134, 1143, 1145, 1147, 1149, 1150, 1164, 1166, 1175, 1182, 1183, 1192, 1194, 1196, 1197, 1199, 1201, 1203,1210, 1217, 1218, 1222, 1231, 1253, 1255, 1257, 1259, 1261, 1264, 1273, 1275, 1276, 1282, 1289, 1296, 1309, 1313, 1315, 1318, 1324, 1325, 1331, 1332, 1334, 1337, 1341, 1344, 1346, 1350, 1352, 1353, 1355, 1360, 1364, 1365, 1366, 1373, 1374, 1381, 1387, 1393, 1397, 1402, 1406, 1415, 1416, 1422, 1423, 1428, 1435, 1439, 1441, 1442, 1443, 1444, 1448, 1449, 1450, 1455, 1457, 1474, 1476, 1484, 1485, 1488, 1493, 1495, 1506, 1511, 1516, 1520, 1521, 1523, 1527, 1530, 1535, 1537, 1540, 1541, 1548, 1549, 1560, 1561, 1562, 1563, 1567, 1569, 1574, 1582, 1584, 1586, 1591, 1593, 1595, 1597, 1610, 1612, 1617, 1621, 1625, 1631, 1633, 1640, 1645, 1647, 1651, 1652, 1658, 1659, 1660, 1666, 1667, 1670, 1673, 1677, 1679, 1686, 1691, 1693, 1696, 1701, 1702, 1708, 1709, 1710, 1712, 1717, 1719, 1721, 1724, 1730, 1747, 1749, 1751, 1754, 1758, 1763, 1768, 1771, 1789, 1796, 1801, 1805, 1807, 1810, 1812, 1815, 1819, 1826, 1827, 1829, 1833, 1838, 1842, 1848, 1849, 1854, 1862, 1864, 1880, 1883, 1887, 1896, 1903, 1910, 1924, 1932, 1933, 1940, 1953, 1957, 1961, 1964,1971, 1974, 1975, 1978, 1983, 1987, 1994, 1995, 1997, 2001, 2011, 2016, 2017, 2023, 2030, 2034, 2044, 2050, 2055, 2057, 2059, 2065, 2066, 2069, 2074, 2076, 2080, 2081, 2086, 2087, 2092, 2106, 2108, 2111, 2113, 2118, 2130, 2137, 2139, 2141, 2143, 2146, 2158, 2160, 2163, 2164, 2169, 2171, 2172, 2177, 2184, 2192, 2195, 2198, 2202, 2204, 2205, 2206, 2228, 2234, 2237, 2239, 2242, 2248, 2256, 2260, 2261, 2263, 2265, 2274, 2276, 2277, 2281, 2291, 2297, 2302, 2304, 2310, 2314, 2319, 2325, 2340, 2346, 2354, 2360, 2361, 2365, 2373, 2374, 2381, 2386, 2388, 2391, 2398, 2403, 2410, 2412, 2423, 2430, 2431, 2436, 2438, 2443, 2447, 2451, 2452, 2456, 2457, 2464, 2465, 2466, 2471, 2472, 2477, 2479, 2484, 2485, 2492, 2494, 2496, 2498, 2499, ...
7	2, 3, 5, 6, 13, 14, 15, 17, 22, 26, 29, 31, 38, 40, 46, 50, 56, 60, 61, 66, 68, 72, 73, 80, 85, 87, 89, 93, 95, 115, 120, 122, 126, 128, 134, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 246, 251, 264, 266, 267, 280, 282, 290, 294, 300, 303, 309, 320, 330, 341, 349, 350, 351, 353, 363, 381, 395, 399, 404, 405, 414, 417, 418, 421, 436, 438, 447, 450, 461, 464, 466, 470, 478, 505, 519, 523, 531, 533, 539, 548, 560, 583, 584, 589, 591, 599, 609, 611, 622, 638, 646, 647, 655, 657, 660, 681, 698, 700, 701, 710, 717, 734, 760, 765, 776, 798, 800, 802, 805, 820, 822, 842, 855, 856, 862, 863, 869, 870, 878, 899, 911, 912, 913, 926, 927, 931, 940, 941, 942, 947, 959, 984, 998, 1016, 1023, 1029, 1046, 1052, 1055, 1081, 1084, 1087, 1111, 1115, 1117, 1131, 1136, 1144, 1148, 1161, 1169, 1170, 1188, 1189, 1194, 1195, 1197, 1221, 1224, 1229, 1248, 1266, 1277, 1280, 1281, 1284, 1291, 1297, 1309, 1316, 1333, 1339, 1348, 1363, 1364, 1371, 1377, 1382, 1402, 1404, 1409, 1418, 1419, 1424, 1425, 1436, 1438, 1439, 1455, 1469, 1471, 1480, 1482, 1489, 1494, 1512, 1513, 1520, 1529, 1537, 1547, 1550, 1554, 1558, 1563, 1565, 1566, 1567, 1572, 1579, 1585, 1596, 1604, 1605, 1606, 1608, 1609, 1622, 1627, 1630, 1651, 1668, 1672, 1679, 1687, 1700, 1708, 1710, 1714, 1716, 1738, 1750, 1753, 1757, 1758, 1762, 1766, 1772, 1783, 1790, 1796, 1797, 1800, 1802, 1808, 1816, 1831,1835, 1839, 1840, 1854, 1855, 1885, 1888, 1899, 1904, 1911, 1915, 1918, 1920, 1929, 1940, 1941, 1958, 1978, 1985, 1990, 1998, 2015, 2039, 2057, 2071, 2072, 2087, 2094, 2097, 2100, 2115, 2126, 2127, 2131, 2139, 2151, 2152, 2156, 2157, 2158, 2159, 2164, 2181, 2186, 2188, 2190, 2193, 2201, 2205, 2219, 2222, 2226, 2230, 2232, 2244, 2245, 2265, 2274, 2293, 2299, 2303, 2306, 2308, 2321, 2359, 2378, 2404, 2408, 2409, 2416, 2436, 2437, 2440, 2450, 2458, 2463, 2466, 2480, 2491, 2493, 2496, ...
8	2, 3, 4, 5, 6, 7, 11, 13, 16, 17, 20, 21, 23, 24, 28, 29, 34, 35, 39, 46, 48, 54, 56, 57, 61, 65, 71, 73, 74, 80, 81, 82, 88, 90, 103, 105, 106, 113, 115, 118, 119, 129, 132, 140, 142, 153, 154, 160, 164, 165, 169, 171, 174, 180, 194, 198, 199, 203, 204, 205, 210, 220, 228, 238, 242, 248, 251, 254, 259, 266, 267, 272, 275, 276, 278, 288, 296, 309,312, 313, 317, 320, 328, 333, 334, 337, 339, 340, 352, 353, 363, 364, 374, 403, 405, 414, 415, 419, 430, 431, 436, 442, 445, 449, 453, 455, 463, 466, 471, 472, 476, 477, 479, 487, 488, 492, 494, 498, 503, 504, 505, 513, 516, 517, 523, 526, 537, 539, 540, 543, 550, 551, 554, 556, 561, 566, 567, 568, 573, 579, 582, 584, 600, 605, 607, 613, 616, 623,624, 628, 639, 643, 649, 656, 657, 677, 681, 690, 701, 702, 703, 710, 713, 719, 721, 725, 730, 732, 738, 742, 745, 748, 758, 760, 761, 768, 769, 772, 778, 786, 788, 795, 798, 800, 805, 810, 811, 819, 821, 829, 833, 843, 845, 855, 856, 857, 874, 879, 883, 891, 894, 895, 912, 913, 914, 917, 919, 928, 930, 931, 936, 952, 962, 963, 965, 966, 986, 992,996, 997, 1007, 1013, 1015, 1019, 1038, 1041, 1042, 1047, 1049, 1053, 1057, 1071, 1072, 1076, 1087, 1088, 1089, 1101, 1119, 1123, 1125, 1126, 1132, 1135, 1136, 1142, 1144, 1150, 1152, 1163, 1170, 1173, 1180, 1183, 1200, 1202, 1218, 1236, 1238, 1246, 1252, 1253, 1259, 1269, 1270, 1275, 1280, 1302, 1305, 1322, 1327, 1330, 1333, 1344, 1353, 1355, 1371, 1381, 1382, 1383, 1388, 1389, 1401, 1404, 1406, 1428, 1434, 1441, 1442, 1446, 1457, 1458, 1463, 1472, 1483, 1485, 1486, 1493, 1496, 1527, 1529, 1533, 1536, 1540, 1542, 1547, 1552, 1554, 1557, 1558, 1567, 1568, 1569, 1571, 1586, 1594, 1598, 1610, 1612, 1618, 1622, 1635, 1638, 1644, 1646, 1650, 1652, 1655, 1666, 1680, 1683, 1687, 1688, 1700, 1706,1710, 1718, 1722, 1737, 1738, 1745, 1751, 1754, 1755, 1757, 1765, 1772, 1788, 1789, 1806, 1809, 1813, 1820, 1823, 1824, 1829, 1836, 1841, 1849, 1850, 1853, 1854, 1864, 1865, 1870, 1892, 1905, 1909, 1910, 1915, 1916, 1926, 1932, 1934, 1935, 1942, 1944, 1945, 1948, 1952, 1956, 1962, 1967, 1972, 1977, 1978, 1985, 1986, 1994, 2001, 2007, 2011, 2013, 2037, 2039, 2040, 2044, 2046, 2051, 2058, 2061, 2068, 2069, 2071, 2073, 2074, 2079, 2097, 2102, 2104, 2108, 2125, 2126, 2131, 2141, 2142, 2143, 2145, 2149, 2152, 2158, 2162, 2163, 2175, 2192, 2193, 2204, 2205, 2206, 2209, 2215, 2222, 2223, 2224, 2226, 2238, 2240, 2245, 2247, 2250, 2254, 2255, 2256, 2260, 2261, 2266, 2279, 2283, 2292, 2296, 2305, 2311, 2312, 2315, 2322, 2333, 2336, 2341, 2343, 2349, 2350, 2360, 2363, 2368, 2369, 2373, 2374, 2376, 2384, 2397, 2400, 2403, 2408, 2414, 2421, 2428, 2432, 2436, 2438, 2441, 2443, 2445, 2447, 2465, 2478, 2479, 2481, 2482, 2486, 2488, 2493, ...
9	2, 3, 8, 10, 11, 13, 20, 21, 26, 30, 50, 51, 56, 60, 67, 76, 78, 88, 91, 98, 102, 117, 129, 134, 146, 154, 159, 163, 171, 186, 189, 190, 191, 193, 198, 200, 209, 210, 212, 222, 240, 249, 250, 265, 267, 269, 278, 279, 299, 300, 307, 333, 344, 363, 383, 388, 390, 398, 399, 406, 425, 429, 436, 438, 439, 444, 450, 458, 496, 504, 509, 520, 523, 533, 540, 544, 570, 580, 585, 603, 634, 637, 646, 658, 666, 668, 691, 698, 704, 716, 721, 735, 748, 798, 806, 809, 818, 827, 844, 866, 875, 886, 887, 889, 894, 939, 952, 962, 984, 989, 998, 1001, 1002, 1003, 1021, 1025, 1029, 1040, 1041, 1044, 1064, 1072, 1075, 1102, 1121, 1139, 1152, 1153, 1186, 1188, 1198, 1207, 1224, 1227, 1234, 1235, 1247, 1264, 1268,1276, 1280, 1285, 1286, 1287, 1300, 1305, 1314, 1331, 1349, 1364, 1399, 1400, 1408, 1409, 1420, 1427, 1432, 1445, 1447, 1473, 1484, 1512, 1519, 1527, 1557, 1560, 1569, 1577, 1590, 1591, 1609, 1614, 1618, 1626, 1630, 1641, 1649, 1654, 1664, 1682, 1685, 1687, 1692, 1704, 1706, 1717, 1739, 1741, 1761, 1769, 1774, 1787, 1805, 1806, 1813, 1826, 1843, 1850, 1865, 1870, 1873, 1907, 1910, 1915, 1937, 1949, 1950, 1959, 1976, 1984, 1990, 2016, 2021, 2029, 2040, 2059, 2062, 2065, 2073, 2082, 2085, 2112, 2124, 2136, 2139, 2141, 2160, 2167, 2174, 2200, 2203, 2223, 2243, 2257, 2276, 2287, 2288, 2298, 2302, 2307, 2339, 2345, 2350, 2357, 2363, 2390, 2395, 2397, 2405, 2406, 2424, 2428, 2440, 2463, 2466, 2469, 2477, 2480, 2497, ...
10	2, 3, 4, 5, 9, 10, 11, 12, 16, 20, 21, 22, 33, 34, 37, 38, 43, 47, 48, 55, 59, 69, 71, 75, 76, 80, 81, 99, 109, 111, 121, 124, 126, 131, 133, 135, 136, 141, 144, 155, 157, 158, 164, 165, 176, 177, 180, 199, 203, 223, 229, 242, 245, 251, 253, 254, 256, 257, 258, 264, 265, 268, 274, 276, 286, 290, 297, 307, 319, 322, 323, 334, 342, 361, 363, 366, 374, 375, 377, 385, 388, 396, 411, 427, 438, 439, 444, 452, 462, 483, 487, 489, 495, 499, 500, 505, 506, 507, 509, 510, 528, 531, 539, 549, 551, 555, 583, 584, 588, 603, 606, 609, 610, 628, 639, 641, 642, 652, 654, 658, 664, 676, 693, 702, 704, 707, 714, 716, 719, 738, 741, 749, 758, 760, 768, 774, 782, 791, 792, 812, 813, 815, 836, 839, 850, 861, 874, 894, 901, 912, 924, 935, 949, 960, 973, 978, 980, 982, 994, 1004, 1012, 1023, 1026, 1035, 1039, 1045, 1054, 1057, 1065, 1066, 1068, 1071, 1087, 1093, 1094, 1105, 1109, 1116, 1127, 1137, 1148, 1155, 1156, 1158, 1160, 1165, 1166, 1178, 1187, 1191, 1199, 1204, 1215, 1219, 1230, 1243, 1244, 1254, 1259, 1265, 1266, 1270, 1276, 1279, 1281, 1291, 1310,1320, 1341, 1342, 1353, 1364, 1365, 1368, 1374, 1375, 1401, 1433, 1452, 1462, 1473, 1474, 1475, 1488, 1496, 1506, 1530, 1539, 1541, 1550, 1551, 1554, 1560, 1567, 1574, 1585, 1593, 1594, 1598, 1605, 1609, 1611, 1615, 1618, 1621, 1637, 1643, 1649, 1659, 1660, 1661, 1676, 1686, 1692, 1695, 1698, 1715, 1716, 1717, 1719, 1738, 1752, 1754, 1761, 1775, 1780, 1791, 1805, 1818, 1819, 1831, 1838, 1840, 1842, 1846, 1848, 1864, 1869, 1873, 1874, 1884, 1901, 1903, 1919, 1923, 1926, 1928, 1929, 1935, 1940, 1946, 1947, 1948, 1962, 1970, 1974, 1990, 1996, 2017, 2023, 2025, 2046, 2058, 2066, 2080, 2099, 2102, 2111, 2112, 2124, 2134, 2143, 2160, 2165, 2170, 2189, 2192, 2210, 2222, 2227, 2232, 2233, 2256, 2258, 2264, 2265, 2271, 2275, 2293, 2303, 2304, 2314, 2320, 2324, 2326, 2330, 2332, 2333, 2335, 2343, 2353, 2366, 2386, 2390, 2392, 2419, 2440, 2457, 2463, 2469, 2473, 2484, ...
11	5, 17, 20, 21, 23, 30, 53, 60, 86, 137, 172, 195, 212, 224, 229, 254, 258, 268, 272, 287, 298, 319, 339, 355, 365, 366, 389, 390, 398, 414, 467, 474, 480, 504, 534, 539, 543, 562, 567, 592, 619, 626, 654, 672, 709, 735, 756, 766, 770, 778, 787, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1091, 1103, 1167, 1194, 1210, 1213, 1239, 1243, 1249, 1264, 1309, 1311, 1341, 1348, 1357, 1377, 1425, 1433, 1437, 1448, 1449, 1459, 1486, 1510, 1528, 1575, 1586, 1601, 1631, 1661, 1696, 1698, 1701, 1748, 1770, 1840, 1859, 1860, 1864, 1874, 1910, 1924, 1926, 1937, 1942, 1954, 1977, 1985, 1993, 2000, 2029, 2039, 2087, 2089, 2123, 2149, 2167, 2215, 2223, 2228, 2238, 2261, 2274, 2294, 2301, 2307, 2311, 2319, 2320, 2323, 2338, 2388, 2393, 2403, 2416, 2438, 2457, 2458, 2461, ...
12	2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311, 313, 320, 324, 328, 329, 339, 343, 346, 348, 350, 363, 374, 376, 390, 407, 412, 413, 417, 424, 426, 429, 434, 437, 438, 443, 447, 455, 463, 468, 480, 482, 485, 494, 502, 503, 507, 511, 515, 525, 533, 534, 545, 549, 555, 558, 559, 568, 571, 582, 593, 595, 599, 610, 612, 614, 619, 625, 632, 638, 640, 641, 647, 653, 659, 676, 684, 686, 697, 699, 701, 702, 707, 710, 716, 718, 724, 725, 729, 738, 742, 749, 767, 768, 777, 780, 793, 794, 797, 798, 801, 805, 811, 820, 832, 836, 840, 842, 853, 854, 861, 862, 868, 871, 872, 876, 893, 905, 909, 915, 923, 924, 949, 953, 958, 959, 961, 963, 967, 972, 974, 978, 988, 1000, 1009, 1015, 1018, 1026, 1031, 1036, 1039, 1040, 1045, 1063, 1067, 1070, 1074, 1080, 1101, 1105, 1110, 1113, 1122, 1123, 1126, 1132, 1134, 1135, 1147, 1156, 1174, 1195, 1196, 1205, 1208, 1210, 1212, 1214, 1219, 1232, 1239, 1251, 1256, 1258, 1261, 1275, 1284, 1299, 1305, 1308, 1325, 1326, 1331, 1336, 1344, 1360, 1365, 1369, 1370, 1373, 1375, 1379, 1394, 1395, 1405, 1413, 1416, 1421, 1426, 1434, 1440, 1452, 1459, 1460, 1464, 1473, 1483, 1486, 1490, 1491, 1492, 1499, 1505, 1516, 1517, 1521, 1524, 1538, 1550, 1552, 1555, 1563, 1569, 1572, 1576, 1589, 1590, 1596, 1598, 1600, 1611, 1616, 1624, 1625, 1635, 1641, 1647, 1652, 1654, 1659, 1660, 1661, 1663, 1669, 1680, 1689, 1690, 1693, 1695, 1700, 1704, 1712, 1713, 1717, 1719, 1726, 1733, 1739, 1741, 1746, 1751, 1756, 1758, 1759, 1764, 1773, 1778, 1780, 1789, 1793, 1811, 1823,1829, 1830, 1832, 1841, 1854, 1856, 1860, 1863, 1875, 1885, 1886, 1889, 1890, 1893, 1897, 1898, 1899, 1902, 1914, 1915, 1927, 1928, 1929, 1933, 1936, 1949, 1951, 1963, 1967, 1971, 1980, 1986, 1989, 2005, 2011, 2018, 2033, 2037, 2040, 2054, 2055, 2062, 2072, 2075, 2076, 2083, 2084, 2089, 2090, 2096, 2102, 2105, 2107, 2111, 2116, 2119, 2122, 2135, 2144, 2153, 2155, 2168, 2183, 2189, 2211, 2224, 2232, 2235, 2241, 2250, 2252, 2257, 2258, 2259, 2261, 2270, 2285, 2300, 2301, 2309, 2313, 2319, 2331, 2337, 2340, 2341, 2356, 2363, 2369, 2371, 2384, 2387, 2392, 2395, 2396, 2401, 2426, 2440, 2443, 2453, 2454, 2457, 2486, 2488, 2497, 2500, ...
13	2, 3, 5, 7, 34, 37, 43, 59, 72, 92, 94, 98, 110, 118, 133, 149, 151, 159, 190, 207, 219, 221, 251, 260, 264, 267, 274, 282, 286, 291, 319, 355, 363, 373, 382, 397, 398, 402, 406, 408, 412, 436, 442, 443, 486, 489, 507, 542, 544, 552, 553, 573, 582, 585, 592, 603, 610, 614, 634, 643, 645, 677, 689, 708, 720, 730, 744, 769, 772, 806, 851, 853, 862, 882, 912, 928, 930, 952, 968, 993, 1006, 1019, 1025, 1040, 1047, 1067, 1097, 1098, 1120, 1125, 1131, 1133, 1153, 1165, 1172, 1175, 1191, 1197, 1236, 1245, 1248, 1262, 1271, 1273, 1275, 1281, 1299, 1305, 1306, 1311, 1346, 1376, 1378, 1400, 1403, 1404, 1412, 1418, 1428, 1431, 1442, 1450, 1471, 1496, 1502, 1570, 1585, 1587, 1660, 1662, 1666, 1736, 1754, 1757, 1836, 1853, 1860, 1861, 1885, 1903, 1913, 1915, 1972, 1975, 1987, 1988, 1995, 2006, 2033, 2052, 2073, 2098, 2191, 2193, 2206, 2210, 2224, 2252, 2255, 2267, 2280, 2302, 2314, 2319, 2324, 2346, 2362, 2366, 2417, 2444, 2450, 2479, 2486, ...
14	2, 3, 10, 11, 14, 15, 16, 17, 18, 21, 24, 25, 29, 37, 43, 44, 46, 48, 49, 52, 54, 61, 66, 69, 72, 73, 78, 84, 86, 87, 99, 101, 104, 106, 114, 115, 128, 133, 135, 136, 143, 145, 146, 148, 164, 169, 170, 173, 200, 219, 224, 226, 228, 231, 234, 240, 244, 248, 255, 262, 275, 281, 282, 298, 301, 305, 320, 321, 322, 327, 330, 339, 347, 359, 363, 368, 369, 384, 385, 387, 401, 417, 433, 435, 442, 450, 455, 458, 482, 488, 494, 496, 508, 517, 519, 521, 553, 559, 566, 579, 582, 590, 601, 616, 621, 625, 628, 630, 636, 638, 654, 657, 669, 675, 691, 692, 694, 711, 712, 714, 719, 724, 725, 728, 732, 737, 750, 756, 757, 765, 768, 773, 774, 778, 780, 793, 797, 803, 810, 827, 828, 831, 842, 848, 860, 865, 869, 895, 901, 910, 916, 919, 922, 923, 926, 931, 936, 938, 951, 960, 967, 971, 972, 982, 1002, 1010, 1022, 1032, 1060, 1063, 1070, 1073, 1074, 1089, 1098, 1105, 1109, 1110, 1123, 1128, 1130, 1131, 1132, 1138, 1147, 1160, 1178, 1179, 1184, 1189, 1190, 1207, 1216, 1218, 1220, 1221, 1291, 1293, 1302, 1303, 1308, 1332, 1349, 1350, 1353, 1357, 1361, 1362,1366, 1373, 1374, 1386, 1394, 1407, 1435, 1448, 1474, 1487, 1511, 1515, 1524, 1535, 1538, 1558, 1562, 1563, 1564, 1581, 1597, 1605, 1614, 1615, 1616, 1626, 1638, 1639, 1645, 1648, 1653, 1654, 1671, 1681, 1683, 1693, 1696, 1698, 1705, 1707, 1710, 1713, 1719, 1723, 1726, 1731, 1739, 1748, 1758, 1768, 1781, 1787, 1789, 1793, 1799, 1800, 1810, 1817, 1823, 1830, 1835, 1839, 1848, 1853, 1859, 1879, 1909, 1929, 1932, 1935, 1961, 1964, 1969, 1979, 1996, 1997, 2002, 2009, 2013, 2015, 2026, 2037, 2056, 2058, 2071, 2074, 2075, 2077, 2080, 2088, 2099, 2104, 2106, 2111, 2133, 2140, 2145, 2147, 2162, 2174, 2175, 2190, 2196, 2204, 2207, 2216, 2219, 2221, 2233, 2234, 2236, 2248, 2251, 2257, 2269, 2273, 2277, 2285, 2286, 2294, 2305, 2309, 2315, 2316, 2319, 2322, 2334, 2339, 2356, 2372, 2378, 2379, 2393, 2403, 2406, 2414, 2418, 2419, 2424, 2425, 2427, 2432, 2434, 2446, 2460, 2467, 2479, 2480, 2482, 2489, 2494, ...
15	2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425, 426, 446, 461,463, 466, 470, 480, 482, 486, 487, 488, 494, 495, 498, 499, 508, 518, 521, 533, 540, 552, 554, 573, 580, 592, 593, 594, 595, 612, 624, 633, 636, 644, 647, 672, 681, 703, 707, 711, 712, 718, 719, 721, 724, 737, 740, 756, 760, 766, 791, 792, 801, 811, 821, 832, 852, 864, 872, 905, 914, 915, 938, 942, 946, 983, 984, 985, 990, 996, 1016, 1025, 1038, 1046, 1056, 1070, 1078, 1085, 1116, 1127, 1129, 1143, 1146, 1178, 1179, 1209, 1213, 1217, 1257, 1261, 1265, 1266, 1271, 1283, 1288, 1307, 1331, 1339, 1344, 1360, 1385, 1422, 1434, 1441, 1448, 1451, 1484, 1491, 1499, 1501, 1505, 1509, 1519, 1522, 1549, 1555, 1574, 1575, 1576, 1585, 1587, 1592, 1605, 1617, 1625, 1639, 1675, 1695, 1698, 1700, 1701, 1705, 1731, 1751, 1753, 1761, 1763, 1772, 1797, 1801, 1834, 1860, 1881, 1893, 1904, 1908, 1923, 1935, 1937, 1938, 1943, 1944, 1956, 1959, 1983, 1985, 2007, 2011, 2014, 2019, 2027, 2032, 2051, 2052, 2061, 2063, 2075, 2078, 2083, 2085, 2106, 2113, 2121, 2123, 2139, 2163, 2168, 2172, 2200, 2206, 2214, 2217, 2226, 2230, 2234, 2248, 2259, 2276, 2278, 2309,2318, 2326, 2328, 2347, 2357, 2373, 2383, 2388, 2398, 2413, 2440, 2460, 2461, 2473, 2476, 2486, ...
16	2, 4, 9, 13, 33, 43, 47, 51, 53, 69, 81, 118, 132, 140, 145, 152, 185, 205, 208, 237, 239, 240, 242, 288, 290, 305, 306, 323, 341, 365, 373, 378, 392, 395, 409, 426, 433, 434, 442, 451, 455, 491, 501, 508, 510, 519, 531, 540, 542, 553, 557, 562, 565, 577, 596, 610, 664, 680, 682, 705, 723, 732, 747, 782, 795, 800, 808, 835, 841, 859, 866, 876, 884,892, 916, 918, 934, 951, 956, 973, 990, 1021, 1022, 1028, 1037, 1053, 1054, 1103, 1106, 1113, 1120, 1121, 1135, 1143, 1174, 1199, 1205, 1209, 1211, 1224, 1232, 1239, 1256, 1284, 1307, 1310, 1343, 1351, 1375, 1377, 1379, 1390, 1403, 1446, 1458, 1475, 1477, 1479, 1495, 1505, 1513, 1529, 1531, 1580, 1585, 1602, 1614, 1616, 1623, 1628, 1640, 1681, 1682, 1691, 1698, 1699, 1732, 1749, 1752, 1753, 1755, 1777, 1785, 1800, 1815, 1836, 1837, 1845, 1868, 1871, 1878, 1885, 1930, 1936, 1942, 1947, 1991, 2053, 2061, 2078, 2083, 2090, 2100, 2104, 2117, 2123, 2140, 2159, 2175, 2191, 2193, 2194, 2227, 2257, 2261, 2274, 2279, 2282, 2286, 2291, 2297, 2328, 2330, 2337, 2359, 2362, 2378, 2405, 2429, 2440, 2467,2480, 2490, ...
17	2, 11, 18, 20, 21, 28, 31, 55, 57, 62, 84, 87, 97, 107, 109, 129, 147, 149, 157, 160, 170, 181, 188, 189, 191, 205, 207, 241, 247, 251, 274, 295, 297, 315, 327, 335, 349, 351, 355, 358, 364, 365, 368, 379, 383, 410, 419, 423, 431, 436, 438, 466, 472, 506, 513, 527, 557, 571, 597, 599, 614, 637, 653, 656, 688, 708, 709, 720, 740, 762, 835, 836, 874,974, 976, 980, 982, 986, 987, 1029, 1045, 1054, 1136, 1151, 1171, 1181, 1185, 1208, 1222, 1230, 1236, 1240, 1242, 1280, 1319, 1332, 1367, 1371, 1375, 1392, 1402, 1435, 1447, 1481, 1495, 1496, 1519, 1525, 1530, 1536, 1547, 1556, 1592, 1603, 1607, 1615, 1683, 1692, 1706, 1726, 1775, 1776, 1782, 1793, 1828, 1833, 1846, 1945, 1967, 1988, 1998, 2001, 2002, 2046, 2049, 2064, 2100, 2103, 2108, 2165, 2183, 2192, 2206, 2223, 2249, 2259, 2268, 2295, 2301, 2303, 2312, 2321, 2343, 2351, 2371, 2395, 2406, 2443, 2446, 2466, 2468, 2471, 2492, 2494, ...
18	2, 5, 6, 7, 8, 9, 11, 26, 35, 38, 42, 49, 54, 56, 66, 73, 80, 81, 88, 96, 100, 106, 107, 118, 120, 121, 122, 126, 131, 134, 175, 190, 196, 197, 207, 208, 227, 234, 235, 240, 245, 266, 271, 275, 296, 301, 302, 305, 308, 312, 322, 335, 342, 343, 347, 350, 353, 362, 367, 378, 379, 381, 384, 397, 399, 415, 418, 423, 442, 444, 455, 464, 467, 473, 480, 494, 505, 513, 520, 530, 556, 557, 574, 575, 581, 609, 628, 664, 665, 666, 689, 695, 700, 702, 708, 711, 712, 720, 723, 729, 739, 745, 795, 810, 823, 847, 859, 861, 871, 897, 900, 921, 930, 949, 951, 975, 981, 996, 999, 1034, 1045, 1050, 1072, 1073, 1081, 1089, 1092, 1102, 1108, 1120, 1128, 1132, 1145, 1163, 1165, 1170, 1227, 1235, 1236, 1258, 1265,1273, 1280, 1291, 1293, 1297, 1301, 1317, 1338, 1339, 1366, 1369, 1377, 1380, 1387, 1391, 1393, 1395, 1403, 1418, 1422, 1450, 1455, 1461, 1469, 1479, 1486, 1494, 1507, 1526, 1527, 1532, 1537, 1544, 1546, 1563, 1566, 1570, 1576, 1596, 1600, 1603, 1605, 1620, 1626, 1627, 1634, 1640, 1662, 1665, 1670, 1671, 1684, 1692, 1721, 1735, 1746, 1747, 1768, 1785, 1798, 1803, 1825, 1830, 1833, 1847, 1850, 1881, 1892, 1899, 1904, 1923, 1938, 1942, 1943, 1944, 1976, 1988, 1994, 2006, 2022, 2045, 2050, 2057, 2059, 2061, 2064, 2071, 2083, 2087, 2120, 2125, 2132, 2145, 2172, 2203, 2221, 2253, 2258, 2269, 2306, 2311, 2325, 2337, 2346, 2373, 2384, 2424, 2440, 2449, 2457, ...
19	2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48, 65, 66, 67, 75, 85, 90, 103, 105, 117, 119, 137, 147, 164, 167, 179, 181, 205, 220, 235, 242, 253, 254, 263, 268, 277, 303, 315, 332, 337, 366, 369, 370, 389, 399, 404, 424, 431, 446, 449, 480, 481, 506, 509, 521, 523, 531, 547, 567, 573, 581, 622, 646, 651, 673, 736, 761, 768, 787, 797, 807, 810, 811, 817, 840, 846, 857, 867, 869, 870, 888, 899, 902, 951, 971, 988, 990, 992, 1010, 1023, 1027, 1030, 1041, 1042, 1072, 1102, 1112, 1116, 1143, 1147, 1208, 1218, 1223, 1252, 1256, 1257, 1260, 1291, 1330, 1335, 1340, 1402, 1454, 1502, 1536, 1538, 1594, 1622, 1625, 1631, 1642, 1648, 1693, 1701, 1709, 1726, 1734, 1759, 1767, 1779, 1794, 1809, 1822, 1858, 1939, 1954, 1977, 1988, 1992, 1999, 2024, 2030, 2065, 2067, 2081, 2101, 2135, 2150, 2180, 2193, 2196, 2219, 2239, 2293, 2325, 2330, 2336, 2338, 2358, 2382, 2390, 2402, 2403, 2409, 2457, 2461, 2478, 2480, 2499, ...
20	2, 3, 4, 9, 11, 12, 16, 19, 26, 34, 45, 48, 54, 63, 70, 78, 82, 83, 86, 91, 96, 97, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 153, 167, 168, 171, 173, 181, 182, 187, 190, 192, 195, 215, 227, 228, 229, 257, 260, 276, 298, 299, 305, 309, 311, 314, 319, 334, 339, 352, 362, 369, 373, 375, 378, 414, 420, 425, 433, 444, 462, 467, 470, 477, 478, 479, 482, 483, 485, 506, 518, 519, 532, 534, 540, 548, 550, 552, 564, 570, 575, 591, 593, 601, 605, 606, 619, 626, 631, 642, 646, 649, 659, 660, 673, 684, 714, 719, 731, 734, 735, 749, 757, 791, 804, 832, 837, 842, 851, 876, 884, 885, 890, 899, 913, 926, 928, 937, 946, 949, 955, 961, 962, 965, 972, 987, 1010, 1044, 1053, 1057, 1085, 1095, 1098,1104, 1114, 1118, 1125, 1141, 1162, 1170, 1188, 1201, 1207, 1217, 1218, 1220, 1224, 1229, 1234, 1239, 1262, 1294, 1306, 1322, 1327, 1342, 1349, 1362, 1369, 1384, 1394, 1400, 1401, 1423, 1464, 1469, 1475, 1477, 1490, 1491, 1511, 1513, 1521, 1523, 1531, 1534, 1540, 1544, 1567, 1569, 1571, 1585, 1593, 1611, 1617, 1628, 1630, 1631, 1637, 1640, 1641, 1650, 1665, 1672, 1674, 1677, 1699, 1705, 1706, 1710, 1711, 1719, 1721, 1739, 1744, 1763, 1767, 1770, 1794, 1810, 1815, 1817, 1818, 1820, 1844, 1869, 1879, 1886, 1892, 1898, 1902, 1918, 1920, 1930, 1931, 1937, 1942, 1945, 1961, 1964, 1967, 1971, 1978, 1999, 2002, 2016, 2031, 2038, 2043, 2057, 2062, 2068, 2073, 2075, 2079, 2114, 2116, 2117, 2155, 2174, 2185, 2189, 2191, 2205, 2215, 2221, 2226, 2236, 2237, 2239, 2248, 2259, 2269, 2270, 2306, 2307, 2313, 2315, 2338, 2360, 2372, 2377, 2393, 2395, 2409, 2418, 2422, 2429, 2430, 2437, 2474, 2482, 2487, ...
21	2, 3, 6, 7, 12, 18, 22, 27, 28, 30, 35, 41, 59, 63, 69, 72, 112, 116, 127, 128, 132, 133, 136, 140, 156, 164, 166, 172, 191, 198, 202, 215, 218, 219, 247, 276, 288, 305, 307, 323, 334, 343, 377, 383, 433, 474, 479, 516, 519, 521, 532, 534, 538, 549, 575, 586, 613, 622, 646, 647, 675, 680, 688, 692, 733, 737, 790, 793, 795, 815, 822, 902, 909, 911,915, 947, 952, 966, 996, 1025, 1033, 1034, 1048, 1073, 1075, 1082, 1087, 1093, 1129, 1144, 1163, 1166, 1182, 1188, 1208, 1241, 1251, 1252, 1269, 1274, 1310, 1340, 1363, 1379, 1418, 1424, 1460, 1461, 1463, 1494, 1501, 1503, 1507, 1508, 1510, 1532, 1540, 1553, 1568, 1590, 1595, 1603, 1623, 1627, 1635, 1660, 1682, 1719, 1728, 1749, 1756, 1766, 1790, 1796, 1797, 1826, 1832, 1857, 1879, 1886, 1924, 1929, 1939, 1954, 1974, 1979, 1982, 1983, 1997, 2004, 2012, 2013, 2064, 2075, 2090, 2097, 2099, 2107, 2113, 2118, 2123, 2128, 2134, 2152, 2200, 2205, 2221, 2234, 2243, 2252, 2257, 2262, 2280, 2306, 2320, 2322, 2323, 2376, 2381, 2383, 2410, 2424, 2436, 2450, 2459, 2480, 2483, ...
22	2, 6, 12, 13, 23, 24, 26, 35, 54, 62, 69, 91, 105, 147, 160, 163, 183, 185, 193, 229, 232, 233, 236, 248, 252, 262, 269, 280, 285, 294, 303, 307, 315, 330, 376, 394, 426, 430, 439, 440, 464, 469, 491, 492, 508, 537, 625, 629, 647, 653, 666, 752, 772, 775, 786, 788, 832, 840, 852, 854, 855, 859, 905, 922, 972, 993, 998, 1012, 1041, 1043, 1061, 1076,1082, 1129, 1136, 1154, 1158, 1173, 1177, 1199, 1208, 1220, 1237, 1266, 1277, 1312, 1336, 1338, 1370, 1386, 1405, 1429, 1439, 1442, 1451, 1455, 1508, 1522, 1543, 1553, 1564, 1566, 1626, 1665, 1734, 1775, 1810, 1821, 1829, 1840, 1844, 1908, 1910, 1938, 1963, 1982, 2009, 2017, 2040, 2070, 2083, 2101, 2105, 2118, 2155, 2163, 2170, 2201, 2203, 2224, 2231, 2244, 2329, 2346, 2347, 2373, 2433, 2450, 2456, 2461, 2485, 2486, 2493, 2496, ...
23	10, 40, 82, 113, 127, 141, 170, 257, 275, 287, 295, 315, 344, 373, 442, 468, 599, 609, 634, 646, 663, 671, 710, 819, 834, 857, 884, 894, 904, 992, 997, 1060, 1069, 1077, 1120, 1143, 1190, 1232, 1253, 1261, 1280, 1291, 1347, 1407, 1436, 1448, 1483, 1514, 1570, 1642, 1680, 1684, 1691, 1744, 1749, 1786, 1801, 1816, 1830, 1879, 1891, 1906, 1946, 1968,1994, 2005, 2158, 2200, 2318, 2444, 2459, ...
24	2, 3, 5, 6, 10, 13, 14, 15, 18, 25, 26, 27, 31, 34, 37, 39, 40, 42, 44, 46, 49, 50, 53, 59, 62, 63, 65, 68, 69, 75, 76, 77, 83, 87, 99, 100, 102, 104, 107, 110, 142, 152, 161, 166, 174, 175, 184, 187, 188, 191, 192, 199, 204, 207, 208, 213, 215, 222, 224, 227, 238, 244, 245, 247, 250, 256, 259, 263, 264, 265, 268, 274, 279, 286, 289, 303, 310, 311,312, 314, 327, 332, 337, 339, 350, 353, 363, 366, 376, 380, 396, 409, 411, 434, 438, 439, 451, 466, 467, 486, 499, 501, 503, 506, 508, 520, 549, 550, 551, 555, 575, 579, 581, 588, 593, 611, 612, 615, 616, 619, 623, 630, 639, 643, 645, 646, 649, 653, 658, 661, 663, 672, 673, 679, 680, 690, 694, 714, 716, 718, 730, 731, 741, 743, 753, 759, 761, 764,781, 792, 799, 813, 816, 818, 834, 837, 842, 854, 861, 863, 867, 872, 873, 875, 878, 880, 886, 892, 903, 937, 965, 978, 989, 990, 1034, 1036, 1041, 1051, 1055, 1080, 1081, 1082, 1095, 1096, 1104, 1105, 1108, 1122, 1127, 1129, 1133, 1135, 1136, 1148, 1171, 1193, 1217, 1221, 1222, 1241, 1242, 1247, 1250, 1253, 1254, 1266, 1280, 1281, 1283, 1287, 1292, 1298, 1302, 1308, 1314, 1323, 1330, 1332, 1340, 1355, 1359, 1361, 1373, 1375, 1378, 1379, 1381, 1384, 1389, 1398, 1399, 1411, 1413, 1424, 1445, 1449, 1457, 1461, 1462, 1473, 1483, 1484, 1486, 1487, 1501, 1505, 1521, 1523, 1544, 1557, 1562, 1573, 1583, 1584, 1586, 1590, 1591, 1597, 1610, 1615, 1630, 1638, 1644, 1651, 1652, 1654, 1660, 1664, 1670,1681, 1688, 1690, 1699, 1704, 1705, 1706, 1707, 1708, 1713, 1715, 1723, 1741, 1743, 1752, 1757, 1767, 1771, 1772, 1775, 1778, 1780, 1797, 1798, 1816, 1822, 1829, 1838, 1848, 1853, 1859, 1869, 1871, 1893, 1904, 1934, 1952, 1957, 1971, 1972, 1973, 1974, 1990, 1998, 2003, 2010, 2011, 2018, 2020, 2022, 2026, 2030, 2031, 2040, 2050, 2070, 2078, 2090, 2094, 2102, 2108, 2115, 2123, 2128, 2129, 2142, 2150, 2156, 2192, 2200, 2208, 2213, 2218, 2238, 2239, 2250, 2264, 2268, 2279, 2295, 2301, 2307, 2310, 2314, 2325, 2334, 2335, 2340, 2346, 2348, 2354, 2364, 2376, 2380, 2381, 2387, 2389, 2393, 2400, 2406, 2407, 2415, 2425, 2427, 2444, 2445, 2453, 2462, 2467, 2474, 2487, ...
25	22, 33, 39, 43, 46, 62, 74, 101, 134, 141, 142, 151, 167, 212, 238, 266, 287, 313, 335, 369, 414, 415, 418, 432, 436, 509, 516, 541, 604, 679, 697, 770, 782, 815, 831, 859, 871, 874, 895, 897, 924, 1039, 1048, 1070, 1085, 1096, 1111, 1121, 1134, 1145, 1170, 1177, 1212, 1239, 1240, 1262, 1321, 1339, 1347, 1361, 1364, 1374, 1407, 1413, 1414, 1449, 1473, 1488, 1513, 1537, 1544, 1624, 1627, 1628, 1630, 1665, 1682, 1691, 1718, 1744, 1758, 1766, 1783, 1792, 1822, 1835, 1979, 2026, 2043, 2046, 2050, 2118, 2159, 2160, 2161, 2165, 2180, 2214, 2223, 2235, 2248, 2250, 2256, 2296, 2308, 2312, 2357, 2368, 2400, 2414, 2433, 2444, 2450, 2487, 2496, ...
26	2, 3, 21, 22, 23, 35, 39, 74, 77, 80, 84, 89, 103, 108, 114, 121, 126, 134, 152, 153, 171, 180, 195, 204, 230, 246, 256, 263, 285, 297, 321, 326, 336, 337, 342, 345, 351, 368, 390, 393, 397, 398, 399, 413, 427, 439, 490, 493, 525, 558, 563, 566, 574, 591, 602, 609, 630, 641, 652, 657, 660, 667, 682, 685, 694, 708, 712, 716, 721, 755, 760, 763, 791,792, 797, 813, 828, 839, 850, 862, 882, 898, 999, 1025, 1029, 1030, 1035, 1038, 1049, 1063, 1073, 1076, 1081, 1105, 1111, 1128, 1162, 1176, 1178, 1185, 1216, 1227, 1234, 1238, 1264, 1268, 1274, 1307, 1318, 1324, 1359, 1364, 1369, 1371, 1373, 1377, 1388, 1392, 1445, 1450, 1451, 1461, 1489, 1496, 1529, 1539, 1550, 1555, 1560, 1610, 1612, 1616, 1624,1632, 1689, 1695, 1794, 1805, 1824, 1857, 1873, 1889, 1899, 1930, 1943, 1950, 1955, 1963, 1979, 1985, 1994, 2003, 2005, 2010, 2085, 2103, 2108, 2113, 2122, 2151, 2153, 2172, 2189, 2214, 2228, 2234, 2236, 2257, 2268, 2293, 2307, 2314, 2337, 2373, 2380, 2400, 2417, 2433, 2436, 2443, 2483, 2487, 2492, ...
27	2, 11, 44, 45, 56, 62, 63, 67, 110, 127, 163, 170, 208, 214, 219, 232, 234, 245, 261, 263, 325, 333, 337, 394, 395, 398, 402, 413, 421, 428, 434, 437, 498, 502, 505, 557, 558, 578, 610, 633, 637, 643, 652, 691, 692, 695, 715, 723, 731, 750, 761, 774, 794, 797, 804, 806, 826, 846, 854, 855, 859, 863, 906, 919, 923, 926, 943, 977, 982, 1045, 1046, 1066, 1081, 1085, 1086, 1091, 1094, 1108, 1157, 1164, 1181, 1184, 1197, 1207, 1212, 1217, 1239, 1271, 1282, 1298, 1300, 1303, 1341, 1371, 1378, 1402, 1407, 1458, 1525, 1579, 1596, 1624, 1632, 1679, 1696, 1705, 1725, 1772, 1775, 1804, 1809, 1827, 1828, 1845, 1853, 1867, 1896, 1909, 1912, 1914, 1958, 1960, 1961, 2004, 2005, 2007, 2030, 2061, 2089, 2122, 2131, 2137, 2138, 2145, 2167, 2276, 2316, 2319, 2321, 2322, 2332, 2393, 2394, 2404, 2439, 2458, ...
28	4, 5, 7, 13, 25, 33, 41, 63, 74, 80, 88, 94, 96, 104, 116, 144, 149, 151, 154, 165, 167, 174, 183, 191, 197, 208, 231, 241, 262, 268, 270, 290, 318, 319, 361, 368, 376, 390, 394, 412, 431, 434, 442, 464, 489, 492, 521, 529, 556, 568, 574, 575, 585, 589, 613, 629, 639, 654, 666, 667, 672, 683, 684, 709, 724, 732, 763, 782, 787, 803, 811, 817, 821, 828, 842, 845, 850, 858, 876, 928, 933, 951, 977, 979, 982, 993, 995, 1006, 1020, 1037, 1045, 1056, 1082, 1086, 1089, 1101, 1111, 1135, 1140, 1156, 1166, 1196, 1217, 1218, 1223, 1224, 1246, 1280, 1285, 1306, 1343, 1357, 1368, 1414, 1426, 1428, 1480, 1484, 1509, 1517, 1530, 1536, 1541, 1567, 1571, 1579, 1589, 1594, 1611, 1707, 1712, 1736, 1739, 1744,1757, 1760, 1763, 1770, 1781, 1791, 1802, 1807, 1818, 1844, 1855, 1862, 1891, 1930, 1942, 1960, 1972, 1985, 1996, 2029, 2043, 2060, 2076, 2170, 2180, 2198, 2213, 2217, 2250, 2261, 2271, 2329, 2340, 2343, 2377, 2412, 2423, 2431, 2464, 2465, 2470, 2482, ...
29	6, 40, 65, 70, 114, 151, 221, 229, 268, 283, 398, 451, 460, 519, 554, 587, 627, 628, 659, 687, 699, 859, 884, 915, 943, 974, 986, 1101, 1120, 1159, 1176, 1212, 1223, 1297, 1312, 1322, 1337, 1390, 1409, 1415, 1446, 1477, 1508, 1592, 1625, 1636, 1683, 1691, 1800, 1802, 1803, 1820, 2020, 2126, 2414, 2421, 2429, 2461, ...
30	2, 6, 7, 8, 9, 16, 18, 20, 26, 29, 31, 32, 33, 40, 41, 47, 57, 76, 82, 87, 88, 93, 101, 109, 120, 121, 125, 133, 140, 144, 162, 163, 175, 178, 183, 186, 191, 196, 215, 216, 218, 227, 233, 242, 253, 266, 267, 273, 276, 278, 304, 312, 317, 319, 328, 336, 374, 380, 391, 408, 413, 421, 436, 438, 439, 448, 453, 457, 474, 475, 479, 491, 500, 502, 505, 510, 511, 521, 526, 541, 558, 560, 564, 609, 621, 625, 628, 636, 648, 657, 660, 678, 682, 683, 692, 705, 712, 714, 727, 738, 746, 759, 767, 782, 783, 802, 803, 826, 831, 833, 836, 837, 844, 846, 851, 863, 882, 898, 924, 925, 931, 938, 946, 953, 970, 975, 976, 984, 988, 990, 1011, 1017, 1054, 1091, 1093, 1115, 1118, 1130, 1142, 1145, 1161, 1162, 1182,1187, 1201, 1219, 1227, 1250, 1260, 1278, 1281, 1294, 1338, 1356, 1364, 1370, 1379, 1389, 1396, 1397, 1402, 1409, 1420, 1433, 1441, 1451, 1459, 1461, 1467, 1486, 1488, 1489, 1492, 1511, 1517, 1535, 1537, 1555, 1565, 1595, 1610, 1614, 1640, 1643, 1644, 1673, 1704, 1709, 1710, 1715, 1724, 1731, 1732, 1735, 1751, 1776, 1786, 1794, 1797, 1800, 1812, 1824, 1827, 1833, 1836, 1844, 1847, 1848, 1859, 1861, 1862, 1865, 1891, 1892, 1917, 1920, 1929, 1941, 1950, 1951, 1983, 1984, 1994, 2011, 2012, 2033, 2045, 2046, 2056, 2074, 2082, 2097, 2104, 2105, 2108, 2118, 2135, 2148, 2149, 2172, 2178, 2196, 2199, 2202, 2203, 2205, 2206, 2207, 2251, 2258, 2264, 2272, 2278, 2294, 2304, 2332, 2360, 2365, 2385, 2388, 2394, 2403, 2434, 2446, 2463, 2465, 2467, 2472, 2488, 2494, ...
31	2, 14, 19, 31, 44, 53, 71, 82, 117, 127, 131, 145, 156, 177, 197, 203, 241, 258, 261, 276, 283, 293, 320, 325, 373, 379, 387, 388, 406, 413, 461, 462, 470, 486, 491, 534, 549, 569, 582, 612, 618, 639, 696, 706, 723, 746, 765, 767, 774, 796, 802, 877, 878, 903, 923, 931, 981, 991, 998, 1002, 1008, 1049, 1077, 1113, 1114, 1115, 1138, 1146, 1159, 1177, 1189, 1200, 1229, 1270, 1281, 1312, 1359, 1429, 1486, 1493, 1496, 1558, 1605, 1614, 1626, 1632, 1649, 1673, 1698, 1754, 1790, 1859, 1865, 1894, 1900, 1904, 1918, 1939, 1948, 1973, 1998, 1999, 2022, 2031, 2038, 2055, 2111, 2167, 2169, 2180, 2236, 2300, 2377, 2387, 2388, 2422, 2462, 2473, ...
32	2, 3, 9, 29, 41, 44, 73, 74, 76, 81, 87, 94, 111, 113, 156, 157, 158, 167, 173, 176, 187, 188, 195, 198, 199, 248, 253, 288, 295, 301, 306, 309, 318, 330, 348, 370, 371, 382, 391, 396, 403, 435, 441, 452, 456, 470, 474, 476, 478, 485, 525, 560, 568, 575, 585, 589, 598, 599, 607, 617, 642, 657, 669, 686, 688, 690, 736, 774, 776, 778, 779, 789, 790,830, 832, 834, 846, 900, 905, 916, 946, 955, 956, 969, 972, 982, 984, 995, 1001, 1018, 1023, 1044, 1078, 1083, 1102, 1141, 1147, 1167, 1170, 1176, 1203, 1213, 1214, 1236, 1247, 1252, 1265, 1322, 1349, 1360, 1407, 1408, 1412, 1426, 1457, 1491, 1528, 1535, 1636, 1660, 1674, 1685, 1734, 1747, 1803, 1812, 1822, 1830, 1834, 1835, 1850, 1856, 1860, 1876,1890, 1891, 1943, 1950, 1979, 1984, 2006, 2027, 2031, 2072, 2078, 2084, 2120, 2135, 2152, 2156, 2159, 2208, 2218, 2230, 2236, 2237, 2271, 2284, 2315, 2318, 2352, 2355, 2377, 2392, 2407, 2415, 2438, 2441, 2448, 2462, ...
33	2, 3, 13, 14, 18, 31, 44, 59, 62, 75, 80, 104, 109, 145, 185, 213, 273, 282, 309, 321, 337, 379, 399, 405, 411, 430, 452, 464, 470, 522, 535, 560, 566, 586, 593, 597, 653, 654, 688, 702, 704, 727, 728, 744, 746, 780, 805, 806, 816, 822, 829, 846, 856, 871, 884, 979, 986, 1001, 1014, 1043, 1087, 1102, 1109, 1133, 1185, 1192, 1211, 1324, 1325, 1340,1410, 1422, 1474, 1476, 1494, 1503, 1556, 1636, 1648, 1661, 1672, 1716, 1720, 1721, 1723, 1725, 1806, 1810, 1829, 1927, 1975, 1983, 1989, 2085, 2168, 2174, 2189, 2207, 2210, 2275, 2290, 2328, 2331, 2359, 2360, 2369, 2379, 2397, 2419, 2487, ...
34	2, 7, 13, 17, 19, 41, 48, 58, 59, 66, 84, 86, 129, 133, 139, 143, 146, 149, 166, 167, 231, 268, 270, 299, 305, 322, 328, 359, 387, 397, 408, 469, 523, 527, 534, 541, 553, 555, 569, 577, 582, 583, 600, 608, 634, 662, 664, 667, 672, 673, 709, 714, 720, 725, 733, 746, 747, 759, 776, 802, 808, 822, 860, 870, 877, 892, 896, 902, 911, 962, 970, 975, 985,1034, 1050, 1051, 1070, 1082, 1090, 1109, 1148, 1150, 1168, 1185, 1198, 1200, 1209, 1212, 1233, 1235, 1261, 1281, 1312, 1338, 1339, 1357, 1380, 1382, 1387, 1403, 1411, 1440, 1441, 1459, 1477, 1483, 1497, 1503, 1560, 1561, 1578, 1583, 1689, 1691, 1710, 1751, 1752, 1755, 1789, 1797, 1826, 1883, 1886, 1887, 1901, 1912, 1950, 1955, 1965, 1971, 1986, 2016, 2032, 2038, 2041, 2060, 2071, 2074, 2083, 2093, 2121, 2126, 2139, 2180, 2182, 2193, 2195, 2209, 2247, 2260, 2292, 2294, 2340, 2461, 2493, ...
35	14, 33, 39, 55, 112, 130, 132, 138, 168, 176, 179, 186, 189, 195, 210, 246, 259, 264, 310, 318, 346, 417, 431, 467, 478, 480, 534, 545, 564, 567, 661, 671, 741, 744, 749, 757, 786, 794, 804, 825, 851, 866, 911, 948, 955, 962, 976, 992, 1014, 1033, 1042, 1082, 1109, 1161, 1193, 1220, 1244, 1260, 1268, 1278, 1313, 1414, 1437, 1476, 1563, 1615, 1618,1658, 1687, 1715, 1738, 1741, 1829, 2016, 2019, 2044, 2045, 2085, 2103, 2105, 2110, 2114, 2118, 2150, 2181, 2198, 2215, 2248, 2260, 2262, 2277, 2286, 2328, 2343, 2366, 2394, 2415, 2440, 2445, 2468, ...
36	3, 7, 9, 10, 11, 14, 27, 30, 33, 37, 40, 51, 62, 71, 78, 80, 82, 95, 117, 125, 137, 142, 148, 157, 185, 189, 191, 197, 223, 232, 236, 252, 255, 265, 287, 321, 323, 324, 326, 332, 337, 341, 354, 369, 391, 414, 416, 418, 430, 487, 509, 545, 558, 563, 582, 588, 589, 618, 675, 694, 714, 715, 739, 773, 776, 788, 884, 898, 909, 928, 932, 935, 948, 950, 952, 969, 989, 996, 1008, 1029, 1040, 1042, 1079, 1085, 1094, 1119, 1124, 1136, 1148, 1159, 1183, 1196, 1207, 1211, 1281, 1296, 1377, 1379, 1380, 1474, 1486, 1516, 1517, 1526, 1538, 1554, 1562, 1563, 1566, 1579, 1582, 1599, 1619, 1628, 1649, 1653, 1655, 1657, 1671, 1773, 1792, 1804, 1809, 1822, 1841, 1849, 1864, 1896, 1912, 1913, 1991, 2012, 2046, 2061, 2071, 2112, 2113, 2115, 2143, 2147, 2160, 2180, 2186, 2211, 2248, 2249, 2263, 2268, 2283, 2302, 2315, 2322, 2347, 2372, 2399, 2426, 2436, 2443, ...
37	61, 77, 94, 97, 99, 113, 126, 130, 134, 147, 161, 172, 187, 202, 208, 246, 261, 273, 285, 302, 320, 432, 444, 503, 523, 525, 563, 666, 680, 709, 740, 757, 787, 902, 962, 964, 969, 1013, 1109, 1111, 1165, 1186, 1219, 1351, 1528, 1587, 1660, 1698, 1710, 1754, 1811, 1972, 2070, 2088, 2137, 2145, 2185, 2210, 2295, 2305, 2348, 2365, 2396, 2398, 2439, 2453, 2462, ...
38	2, 10, 13, 18, 24, 28, 43, 47, 49, 54, 65, 83, 98, 106, 143, 152, 177, 184, 190, 194, 195, 208, 241, 249, 259, 264, 292, 315, 319, 345, 353, 355, 386, 394, 481, 500, 517, 525, 534, 535, 550, 556, 595, 601, 649, 656, 680, 686, 687, 697, 707, 710, 756, 798, 804, 817, 818, 829, 839, 841, 858, 864, 891, 906, 912, 932, 948, 973, 991, 994, 1012, 1030, 1032, 1036, 1046, 1057, 1060, 1066, 1071, 1089, 1166, 1175, 1203, 1205, 1227, 1234, 1237, 1261, 1283, 1284, 1319, 1327, 1335, 1336, 1337, 1356, 1372, 1387, 1412, 1413, 1439, 1450, 1460, 1483, 1488, 1495, 1502, 1515, 1518, 1521, 1532, 1538, 1676, 1677, 1685, 1686, 1690, 1720, 1722, 1734, 1739, 1758, 1759, 1762, 1782, 1792, 1814, 1822, 1833, 1834, 1838, 1839, 1842, 1861, 1869, 1884, 1918, 1922, 1972, 1975, 1999, 2005, 2012, 2033, 2042, 2056, 2093, 2119, 2221, 2244, 2251, 2264, 2325, 2356, 2359, 2371, 2381, 2383, 2427, 2436, 2446, 2479, ...
39	10, 23, 29, 86, 240, 254, 284, 308, 340, 369, 371, 378, 382, 393, 407, 422, 448, 459, 574, 582, 613, 619, 667, 686, 703, 767, 769, 844, 851, 861, 875, 881, 944, 987, 995, 1030, 1056, 1207, 1219, 1233, 1251, 1270, 1279, 1292, 1343, 1372, 1399, 1409, 1445, 1468, 1497, 1500, 1557, 1586, 1598, 1628, 1633, 1645, 1667, 1677, 1712, 1760, 1803, 1807, 1835,1933, 2004, 2014, 2028, 2039, 2057, 2113, 2136, 2143, 2196, 2224, 2229, 2271, 2291, 2376, 2391, 2397, 2398, ...
40	2, 3, 4, 31, 37, 39, 46, 77, 98, 119, 124, 143, 144, 154, 169, 197, 205, 232, 245, 266, 291, 295, 297, 305, 308, 319, 332, 333, 413, 426, 431, 437, 454, 459, 472, 483, 513, 528, 531, 542, 579, 617, 619, 635, 647, 677, 724, 737, 748, 780, 806, 815, 819, 820, 840, 851, 858, 870, 875, 907, 920, 927, 940, 983, 1002, 1028, 1048, 1050, 1064, 1109, 1125,1149, 1179, 1197, 1209, 1222, 1229, 1233, 1244, 1248, 1251, 1320, 1337, 1345, 1410, 1434, 1471, 1522, 1526, 1566, 1574, 1576, 1578, 1607, 1723, 1725, 1738, 1758, 1794, 1877, 1885, 1886, 1968, 1999, 2000, 2009, 2010, 2023, 2041, 2058, 2064, 2073, 2093, 2136, 2141, 2148, 2171, 2182, 2183, 2204, 2215, 2222, 2232, 2273, 2275, 2287, 2317, 2323, 2342, 2355, 2364, 2396, 2423, 2428, 2450, 2478, 2487, ...
41	14, 53, 55, 58, 71, 76, 82, 211, 248, 271, 296, 316, 430, 433, 439, 472, 545, 553, 555, 596, 663, 677, 682, 746, 814, 832, 885, 926, 947, 959, 1048, 1121, 1146, 1175, 1207, 1300, 1302, 1390, 1404, 1445, 1466, 1477, 1541, 1650, 1699, 1702, 1706, 1756, 1757, 1782, 1902, 1944, 1963, 1994, 2012, 2042, 2122, 2163, 2215, 2344, 2348, 2369, 2439, 2442, 2465, ...
42	2, 6, 8, 11, 17, 23, 35, 41, 49, 58, 65, 85, 88, 96, 97, 103, 107, 111, 129, 139, 144, 161, 170, 194, 197, 213, 214, 217, 223, 230, 236, 238, 247, 274, 283, 290, 297, 298, 301, 323, 339, 348, 360, 381, 382, 383, 389, 393, 398, 403, 427, 445, 452, 473, 480, 497, 500, 511, 523, 561, 575, 595, 601, 604, 606, 615, 629, 651, 652, 680, 685, 702, 705, 725, 762, 770, 774, 780, 781, 782, 805, 814, 857, 892, 912, 918, 942, 987, 1004, 1027, 1030, 1039, 1045, 1054, 1099, 1125, 1126, 1131, 1155, 1160, 1168, 1169, 1215, 1231, 1235, 1246, 1249, 1256, 1271, 1279, 1289, 1291, 1314, 1315, 1325, 1373, 1383, 1385, 1386, 1415, 1416, 1420, 1436, 1454, 1460, 1468, 1476, 1498, 1519, 1522, 1556, 1585, 1629, 1657, 1665, 1687, 1692, 1693, 1708, 1713, 1715, 1716, 1729, 1730, 1735, 1744, 1779, 1784, 1787, 1794, 1801, 1808, 1810, 1812, 1816, 1842, 1847, 1851, 1857, 1863, 1914, 1924, 1958, 1966, 1974, 1975, 1991, 2001, 2002, 2013, 2022, 2029, 2036, 2052, 2124, 2132, 2138, 2157, 2182, 2207, 2218, 2228, 2229, 2246, 2251, 2257, 2276, 2279, 2280, 2295, 2303, 2338, 2344, 2360, 2362, 2378, 2387, 2389, 2396, 2406, 2408, 2424, 2465, 2472, 2473, 2486, ...
43	15, 21, 26, 86, 89, 114, 123, 163, 180, 310, 332, 377, 409, 438, 448, 457, 477, 526, 534, 556, 586, 612, 653, 665, 690, 692, 709, 760, 783, 803, 821, 848, 877, 899, 909, 942, 981, 1041, 1042, 1043, 1066, 1068, 1069, 1106, 1126, 1139, 1197, 1310, 1386, 1476, 1561, 1599, 1606, 1636, 1774, 1802, 1835, 1867, 2033, 2068, 2153, 2155, 2229, 2237, 2256, 2269, 2276, 2324, 2467, ...
44	25, 60, 70, 77, 86, 116, 117, 119, 137, 151, 166, 175, 176, 177, 181, 184, 202, 208, 229, 241, 300, 313, 328, 332, 344, 385, 402, 442, 444, 446, 453, 467, 470, 500, 501, 522, 527, 531, 532, 534, 568, 584, 592, 629, 690, 696, 701, 706, 709, 728, 734, 749, 795, 812, 828, 838, 844, 845, 855, 857, 950, 951, 965, 991, 1009, 1011, 1084, 1090, 1098, 1102,1125, 1142, 1168, 1169, 1170, 1179, 1181, 1191, 1212, 1213, 1234, 1246, 1247, 1262, 1284, 1285, 1304, 1310, 1311, 1358, 1368, 1387, 1411, 1421, 1432, 1459, 1468, 1483, 1485, 1515, 1538, 1542, 1570, 1572, 1587, 1616, 1618, 1628, 1629, 1636, 1646, 1647, 1654, 1662, 1676, 1687, 1710, 1723, 1732, 1734, 1750, 1804, 1807, 1831, 1861, 1865, 1875, 1882, 1899, 1921, 1925, 1942, 1945, 1996, 2014, 2015, 2023, 2036, 2066, 2071, 2078, 2088, 2136, 2149, 2165, 2223, 2232, 2262, 2281, 2287, 2435, 2440, 2462, 2463, 2498, ...
45	11, 18, 23, 24, 29, 33, 40, 51, 61, 65, 85, 91, 95, 113, 153, 206, 207, 218, 264, 276, 285, 292, 313, 332, 347, 362, 367, 391, 417, 418, 424, 426, 428, 439, 444, 451, 454, 468, 476, 493, 542, 601, 634, 642, 653, 657, 680, 705, 734, 764, 783, 787, 856, 867, 873, 915, 942, 945, 953, 964, 971, 1013, 1022, 1044, 1054, 1058, 1070, 1111, 1133, 1152, 1242, 1252, 1274, 1304, 1322, 1334, 1365, 1404, 1405, 1406, 1423, 1438, 1442, 1456, 1483, 1500, 1524, 1527, 1531, 1544, 1553, 1562, 1582, 1662, 1671, 1688, 1723, 1736, 1743, 1751, 1761, 1812, 1830, 1833, 1835, 1837, 1910, 1946, 1985, 1989, 2017, 2022, 2026, 2091, 2095, 2129, 2143, 2166, 2170, 2215, 2271, 2272, 2282, 2283, 2288, 2297, 2300, 2305, 2306,2308, 2329, 2336, 2364, 2389, 2402, 2416, 2418, 2443, 2448, 2452, 2466, 2472, 2485, 2496, ...
46	2, 3, 7, 16, 17, 18, 25, 46, 47, 106, 110, 111, 118, 136, 144, 145, 230, 238, 361, 382, 422, 439, 474, 494, 495, 519, 588, 639, 657, 707, 733, 751, 802, 831, 863, 902, 925, 976, 994, 1001, 1059, 1140, 1181, 1240, 1249, 1319, 1375, 1442, 1458, 1508, 1699, 1724, 1734, 1751, 1757, 1760, 1766, 1807, 1849, 1897, 1904, 1914, 1922, 1931, 1978, 2029, 2057,2075, 2139, 2166, 2183, 2209, 2216, 2230, 2241, 2246, 2270, 2307, 2309, 2367, 2429, 2446, 2456, 2460, ...
47	5, 17, 19, 55, 62, 75, 89, 98, 99, 132, 172, 186, 197, 220, 268, 278, 279, 288, 439, 443, 496, 579, 583, 587, 742, 777, 825, 911, 966, 1003, 1029, 1155, 1202, 1267, 1377, 1461, 1472, 1473, 1530, 1562, 1634, 1682, 1705, 1783, 1848, 1852, 1885, 1900, 1943, 1957, 1969, 2036, 2044, 2105, 2111, 2265, 2282, 2485, 2492, ...
48	5, 7, 10, 16, 17, 39, 41, 51, 55, 70, 96, 97, 104, 132, 136, 137, 148, 182, 195, 206, 216, 227, 234, 236, 245, 262, 267, 278, 283, 287, 305, 310, 331, 361, 362, 380, 384, 409, 410, 411, 423, 438, 446, 464, 473, 495, 522, 549, 561, 564, 578, 587, 591, 592, 602, 619, 622, 633, 665, 679, 687, 714, 741, 759, 777, 784, 813, 828, 830, 847, 849, 853, 874,879, 924, 931, 940, 977, 1015, 1030, 1037, 1057, 1058, 1090, 1107, 1141, 1145, 1160, 1170, 1211, 1246, 1265, 1285, 1296, 1299, 1304, 1306, 1312, 1313, 1316, 1328, 1373, 1393, 1400, 1409, 1420, 1443, 1500, 1518, 1542, 1546, 1557, 1570, 1574, 1643, 1644, 1653, 1656, 1685, 1686, 1688, 1727, 1737, 1756, 1759, 1761, 1762, 1763, 1780, 1792, 1807, 1900,1924, 1927, 1930, 1936, 1959, 1964, 1977, 2015, 2018, 2019, 2033, 2041, 2050, 2059, 2070, 2089, 2122, 2133, 2179, 2208, 2217, 2222, 2255, 2302, 2313, 2318, 2342, 2350, 2384, 2407, 2416, 2417, 2430, 2440, 2443, 2470, 2477, 2487, ...
49	2, 5, 50, 57, 62, 67, 104, 123, 138, 151, 155, 186, 194, 233, 236, 253, 261, 269, 306, 337, 359, 416, 425, 439, 557, 613, 695, 705, 742, 748, 757, 821, 832, 884, 885, 892, 897, 902, 936, 937, 988, 1125, 1165, 1176, 1248, 1265, 1320, 1337, 1386, 1401, 1544, 1590, 1594, 1675, 1707, 1712, 1714, 1722, 1725, 1798, 1840, 1844, 2037, 2106, 2117, 2148, 2153, 2154, 2155, 2175, 2192, 2208, 2235, 2286, 2292, 2376, 2384, 2416, 2422, 2470, ...
50	6, 7, 17, 29, 42, 44, 56, 59, 62, 106, 108, 112, 141, 161, 163, 173, 194, 210, 212, 229, 231, 254, 296, 306, 342, 360, 382, 384, 395, 400, 416, 418, 431, 435, 457, 458, 484, 492, 517, 519, 530, 531, 540, 551, 579, 609, 618, 637, 644, 672, 697, 713, 722, 724, 732, 790, 791, 794, 810, 839, 850, 852, 876, 898, 902, 914, 926, 928, 937, 938, 970, 1010,1091, 1118, 1123, 1129, 1150, 1157, 1165, 1182, 1214, 1229, 1247, 1253, 1258, 1269, 1272, 1320, 1321, 1378, 1399, 1402, 1422, 1425, 1474, 1500, 1532, 1539, 1573, 1578, 1584, 1586, 1588, 1622, 1624, 1635, 1651, 1653, 1696, 1699, 1725, 1756, 1811, 1875, 1912, 1948, 1970, 2017, 2076, 2092, 2114, 2115, 2119, 2124, 2131, 2133, 2147, 2175, 2219, 2253, 2260, 2295, 2347, 2406, 2462, 2463, 2464, ...
51	30, 39, 42, 56, 73, 99, 114, 142, 193, 214, 263, 268, 320, 329, 381, 397, 432, 435, 446, 476, 477, 499, 538, 580, 584, 585, 652, 685, 689, 837, 845, 867, 954, 1040, 1108, 1160, 1164, 1283, 1302, 1441, 1450, 1456, 1469, 1609, 1645, 1718, 1720, 1759, 2004, 2065, 2068, 2206, 2289, 2332, 2377, 2383, 2416, 2438, 2443, 2458, ...
52	11, 16, 37, 70, 96, 121, 131, 187, 280, 281, 317, 396, 441, 501, 554, 582, 696, 704, 733, 735, 765, 831, 876, 891, 894, 967, 1013, 1014, 1049, 1071, 1085, 1096, 1119, 1129, 1151, 1162, 1170, 1281, 1289, 1435, 1544, 1594, 1597, 1620, 1653, 1697, 1738, 1740, 1774, 1831, 1864, 1893, 1895, 1899, 1924, 1931, 1944, 1948, 1955, 1977, 2007, 2014, 2021, 2083, 2084, 2186, 2215, 2216, 2268, 2283, 2375, 2422, 2461, ...
53	24, 45, 60, 165, 235, 272, 285, 298, 307, 381, 416, 429, 623, 799, 858, 924, 929, 936, 1034, 1067, 1076, 1130, 1223, 1243, 1327, 1488, 1592, 1594, 1659, 1673, 1676, 1678, 1762, 1873, 1917, 1928, 1997, 2110, 2117, 2141, 2166, 2231, 2243, 2293, 2338, 2380, 2420, 2424, 2428, 2463, 2466, ...
54	2, 7, 9, 15, 23, 24, 25, 42, 62, 63, 78, 91, 122, 151, 159, 201, 208, 222, 248, 252, 277, 281, 284, 286, 291, 299, 316, 340, 342, 378, 395, 440, 445, 460, 478, 498, 522, 528, 532, 558, 580, 600, 668, 688, 701, 710, 726, 743, 752, 757, 763, 766, 771, 778, 787, 796, 798, 805, 812, 856, 864, 882, 893, 895, 963, 981, 987, 992, 1014, 1031, 1046, 1061, 1088, 1098, 1108, 1128, 1163, 1170, 1182, 1191, 1201, 1234, 1266, 1269, 1343, 1373, 1383, 1386, 1460, 1467, 1487, 1508, 1535, 1544, 1575, 1588, 1598, 1650, 1662, 1668, 1697, 1717, 1730, 1738, 1750, 1762, 1766, 1795, 1847, 1852, 1869, 1878, 1894, 1905, 1910, 1912, 1925, 1932, 1952, 1963, 1996, 2015, 2018, 2021, 2105, 2108, 2122, 2141, 2161, 2170, 2188, 2194, 2203, 2235, 2238, 2245, 2251, 2266, 2335, 2366, 2379, 2403, 2431, 2496, ...
55	7, 14, 39, 53, 61, 83, 127, 148, 154, 199, 208, 228, 249, 281, 288, 340, 430, 435, 476, 511, 525, 533, 534, 541, 594, 663, 668, 670, 671, 678, 683, 728, 739, 821, 830, 865, 867, 871, 873, 891, 922, 982, 992, 1037, 1038, 1059, 1090, 1206, 1212, 1252, 1311, 1413, 1443, 1456, 1522, 1537, 1544, 1575, 1616, 1639, 1651, 1668, 1699, 1735, 1741, 1742, 1754, 1756, 1790, 1806, 1820, 1821, 1823, 1958, 2005, 2058, 2064, 2149, 2167, 2169, 2188, 2194, 2226, 2291, 2316, ...
56	2, 5, 12, 19, 23, 33, 34, 84, 130, 136, 159, 167, 183, 187, 192, 197, 226, 229, 253, 259, 260, 269, 274, 323, 332, 337, 381, 404, 444, 464, 487, 491, 496, 520, 527, 577, 583, 614, 666, 672, 678, 683, 688, 702, 742, 747, 748, 762, 790, 830, 857, 865, 869, 871, 898, 901, 946, 1016, 1163, 1245, 1273, 1275, 1280, 1361, 1362, 1371, 1386, 1410, 1457, 1473, 1476, 1488, 1515, 1537, 1559, 1563, 1567, 1570, 1574, 1581, 1600, 1614, 1647, 1689, 1699, 1725, 1769, 1787, 1796, 1804, 1809, 1811, 1837, 1861, 1883, 1888, 1917, 1940, 1964, 1986, 2006, 2027, 2039, 2053, 2137, 2141, 2155, 2179, 2200, 2221, 2227, 2232, 2241, 2318, 2339, 2341, 2345, 2359, 2377, 2389, 2391, 2396, 2401, ...
57	5, 42, 45, 54, 95, 125, 145, 147, 166, 211, 250, 274, 338, 407, 445, 522, 619, 621, 636, 648, 649, 657, 691, 733, 745, 775, 808, 878, 903, 920, 946, 1040, 1064, 1097, 1105, 1122, 1137, 1167, 1168, 1188, 1190, 1197, 1209, 1210, 1211, 1269, 1280, 1319, 1374, 1443, 1477, 1491, 1514, 1520, 1553, 1565, 1586, 1592, 1596, 1624, 1630, 1639, 1656, 1667, 1703, 1769, 1775, 1778, 1790, 1824, 1900, 1918, 1925, 1935, 1939, 1952, 1996, 2023, 2076, 2151, 2168, 2201, 2258, 2294, 2325, 2331, 2333, 2353, 2369, 2407, 2460, 2487, ...
58	7, 15, 25, 62, 119, 123, 154, 202, 245, 285, 294, 295, 357, 371, 476, 626, 664, 690, 708, 723, 737, 768, 783, 803, 825, 826, 835, 841, 842, 867, 871, 897, 904, 934, 953, 1066, 1069, 1088, 1097, 1108, 1183, 1197, 1202, 1259, 1302, 1364, 1461, 1478, 1497, 1528, 1559, 1638, 1715, 1760, 1764, 1815, 1821, 1849, 1854, 1874, 2047, 2086, 2091, 2100, 2133,2160, 2231, 2242, 2316, 2422, ...
59	19, 70, 102, 116, 126, 188, 209, 257, 294, 359, 451, 461, 468, 470, 638, 653, 710, 762, 766, 781, 824, 901, 939, 964, 995, 1036, 1047, 1098, 1150, 1211, 1234, 1243, 1256, 1278, 1389, 1401, 1422, 1436, 1454, 1492, 1523, 1639, 1703, 1705, 1826, 1913, 1915, 1978, 2013, 2043, 2131, 2140, 2200, 2208, 2228, 2229, 2307, 2309, 2323, 2365, 2480, 2485, ...
60	3, 4, 11, 12, 14, 25, 48, 57, 62, 70, 84, 93, 100, 126, 135, 138, 167, 170, 171, 184, 186, 194, 198, 205, 212, 217, 220, 222, 239, 240, 285, 292, 300, 318, 321, 339, 361, 378, 390, 391, 404, 419, 461, 500, 501, 509, 526, 546, 574, 606, 622, 635, 670, 671, 675, 717, 721, 723, 744, 747, 778, 789, 802, 804, 809, 822, 832, 903, 919, 929, 937, 965, 1001, 1012, 1016, 1024, 1034, 1050, 1084, 1086, 1095, 1101, 1132, 1158, 1164, 1174, 1187, 1211, 1215, 1217, 1223, 1262, 1266, 1276, 1282, 1303, 1320, 1342, 1343, 1346, 1365, 1394, 1395, 1403, 1425, 1430, 1450, 1485, 1502, 1504, 1505, 1506, 1624, 1676, 1705, 1711, 1719, 1722, 1741, 1747, 1748, 1750, 1753, 1766, 1797, 1806, 1859, 1870, 1882, 1900, 1912,1913, 1916, 1925, 1994, 2001, 2004, 2027, 2041, 2085, 2090, 2095, 2107, 2116, 2122, 2126, 2144, 2169, 2177, 2185, 2188, 2199, 2306, 2317, 2331, 2334, 2341, 2366, 2379, 2394, 2406, 2411, 2426, 2429, 2437, 2486, 2488, 2490, 2498, ...
61	2, 19, 69, 88, 138, 155, 205, 234, 306, 336, 420, 425, 455, 470, 525, 555, 561, 608, 626, 667, 674, 766, 779, 846, 851, 937, 971, 998, 1053, 1055, 1112, 1137, 1263, 1329, 1383, 1445, 1448, 1499, 1544, 1551, 1578, 1640, 1689, 1724, 1750, 1793, 1868, 1892, 2143, 2241, 2358, 2407, 2414, 2441, ...
62	2, 6, 10, 36, 65, 74, 78, 83, 106, 115, 120, 148, 154, 161, 163, 168, 176, 189, 194, 197, 266, 270, 288, 331, 385, 399, 407, 410, 412, 413, 431, 468, 513, 524, 546, 569, 572, 578, 581, 600, 611, 619, 625, 626, 647, 719, 723, 756, 832, 834, 849, 922, 986, 1006, 1007, 1047, 1064, 1102, 1199, 1217, 1247, 1332, 1335, 1349, 1353, 1372, 1374, 1382, 1400,1405, 1410, 1418, 1419, 1422, 1426, 1444, 1456, 1457, 1554, 1568, 1590, 1622, 1678, 1680, 1683, 1684, 1695, 1725, 1797, 1840, 1847, 1861, 1874, 1895, 1939, 1950, 1987, 1988, 1990, 1999, 2070, 2081, 2093, 2180, 2183, 2212, 2216, 2220, 2227, 2266, 2303, 2337, 2349, 2389, 2408, 2418, 2426, ...
63	3, 7, 12, 40, 92, 122, 178, 191, 195, 214, 282, 283, 297, 299, 300, 418, 437, 467, 493, 497, 507, 512, 530, 597, 630, 664, 667, 701, 744, 800, 823, 876, 899, 944, 965, 1040, 1043, 1061, 1063, 1070, 1083, 1112, 1141, 1209, 1254, 1322, 1325, 1345, 1378, 1398, 1416, 1430, 1532, 1563, 1625, 1704, 1726, 1728, 1746, 1863, 1875, 1921, 1997, 2030, 2085, 2179, 2220, 2319, 2392, 2461, 2476, ...
64	3, 9, 21, 30, 54, 65, 75, 96, 112, 114, 132, 156, 163, 181, 191, 229, 251, 332, 342, 360, 363, 376, 428, 430, 432, 448, 527, 562, 583, 588, 589, 605, 726, 738, 763, 804, 831, 839, 847, 850, 884, 971, 1068, 1099, 1127, 1142, 1167, 1181, 1198, 1199, 1201, 1225, 1243, 1255, 1273, 1299, 1306, 1315, 1379, 1489, 1540, 1547, 1568, 1596, 1673, 1678, 1713,1714, 1754, 1801, 1812, 1818, 1823, 1847, 1861, 1878, 1906, 1957, 1960, 1962, 1987, 2046, 2069, 2087, 2098, 2109, 2118, 2142, 2177, 2195, 2325, 2330, 2385, 2418, 2434, 2457, ...
65	2, 68, 71, 83, 87, 92, 132, 137, 316, 390, 394, 407, 451, 534, 553, 587, 597, 622, 652, 733, 735, 771, 781, 846, 871, 879, 892, 901, 943, 987, 1047, 1077, 1102, 1120, 1179, 1208, 1219, 1224, 1229, 1261, 1303, 1371, 1614, 1623, 1648, 1726, 1745, 1796, 1810, 1813, 1821, 1842, 1920, 1965, 2147, 2153, 2156, 2193, 2330, 2434, ...
66	9, 38, 40, 53, 62, 105, 107, 125, 129, 134, 156, 183, 193, 210, 276, 283, 294, 306, 307, 317, 345, 371, 388, 399, 447, 504, 508, 514, 529, 542, 585, 594, 648, 668, 671, 676, 717, 719, 722, 723, 743, 752, 772, 784, 818, 833, 840, 844, 870, 880, 896, 941, 942, 964, 978, 1021, 1087, 1115, 1144, 1168, 1182, 1194, 1212, 1215, 1239, 1366, 1378, 1527, 1532, 1562, 1603, 1630, 1644, 1712, 1718, 1780, 1831, 1851, 1864, 1881, 1882, 1885, 1890, 1929, 1941, 1943, 1949, 1957, 1980, 2003, 2036, 2052, 2110, 2124, 2158, 2226, 2269, 2317, 2331, 2344, 2366, 2388, 2422, 2452, ...
67	46, 122, 238, 304, 314, 315, 328, 332, 346, 372, 382, 426, 440, 491, 496, 510, 524, 528, 566, 638, 733, 826, 1016, 1054, 1071, 1214, 1309, 1338, 1388, 1401, 1457, 1512, 1536, 1582, 1624, 1718, 1773, 1814, 1816, 1825, 1952, 1985, 2021, 2072, 2308, 2349, 2449, 2481, ...
68	47, 85, 88, 94, 150, 175, 177, 190, 194, 197, 203, 210, 217, 233, 254, 259, 315, 316, 322, 345, 371, 405, 451, 500, 505, 510, 562, 572, 619, 658, 730, 759, 838, 889, 926, 959, 983, 1041, 1090, 1110, 1136, 1145, 1218, 1249, 1251, 1336, 1373, 1374, 1376, 1392, 1416, 1418, 1425, 1484, 1513, 1547, 1562, 1567, 1581, 1590, 1641, 1657, 1804, 1824, 1829, 1903, 1949, 1952, 1966, 1971, 1990, 2001, 2028, 2031, 2059, 2095, 2109, 2119, 2277, 2372, 2381, 2416, 2421, ...
69	2, 22, 26, 43, 82, 218, 271, 273, 386, 392, 412, 469, 589, 599, 616, 686, 689, 807, 842, 848, 855, 867, 1055, 1138, 1170, 1251, 1287, 1352, 1393, 1422, 1555, 1590, 1617, 1662, 1712, 1745, 1851, 1998, 1999, 2069, 2076, 2189, 2213, 2279, 2328, 2440, ...
70	3, 12, 16, 19, 24, 37, 50, 57, 110, 148, 166, 169, 174, 181, 192, 212, 228, 233, 242, 330, 370, 380, 469, 492, 505, 514, 576, 583, 655, 657, 676, 678, 716, 726, 734, 776, 786, 829, 857, 994, 1040, 1099, 1114, 1154, 1181, 1186, 1244, 1338, 1348, 1350, 1354, 1379, 1397, 1436, 1444, 1509, 1517, 1549, 1557, 1591, 1645, 1741, 1752, 1755, 1801, 1813, 1855, 1900, 1925, 1935, 1949, 1960, 1971, 2003, 2004, 2069, 2082, 2084, 2089, 2113, 2130, 2167, 2259, 2286, 2288, 2301, 2338, 2361, 2362, 2380, 2382, 2386, 2393, 2401, 2420, ...
71	3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981, 1033, 1067, 1170, 1195, 1253, 1284, 1349, 1385, 1409, 1479, 1709, 1724, 1726, 1735, 1875, 1950, 1984, 2012, 2070, 2124, 2133, 2161, 2194, 2202, 2424, 2432, 2459, ...
72	3, 18, 36, 43, 51, 77, 89, 150, 178, 195, 209, 219, 244, 254, 265, 280, 290, 294, 314, 322, 340, 341, 362, 363, 371, 377, 386, 416, 450, 474, 488, 494, 511, 546, 548, 554, 576, 611, 625, 647, 723, 740, 760, 766, 800, 801, 803, 807, 820, 827, 835, 883, 940, 943, 956, 976, 1013, 1041, 1105, 1161, 1193, 1198, 1217, 1257, 1321, 1333, 1379, 1380, 1405,1430, 1472, 1550, 1555, 1568, 1698, 1729, 1736, 1750, 1828, 1829, 1841, 1844, 1950, 1952, 1954, 1994, 1998, 2060, 2074, 2123, 2129, 2160, 2174, 2225, 2233, 2244, 2284, 2286, 2392, 2401, 2407, 2411, 2417, 2428, 2473, 2488, ...
73	11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966, 1021, 1112, 1203, 1334, 1433, 1485, 1724, 1822, 1959, 1998, 2115, 2154, 2432, 2465, 2486, ...
74	16, 19, 21, 49, 56, 63, 71, 74, 77, 83, 92, 96, 99, 160, 172, 197, 198, 230, 241, 280, 283, 415, 425, 448, 490, 520, 627, 691, 735, 784, 803, 829, 842, 853, 871, 872, 893, 894, 973, 981, 989, 1043, 1060, 1061, 1071, 1179, 1182, 1203, 1290, 1299, 1317, 1370, 1389, 1477, 1478, 1496, 1562, 1601, 1624, 1714, 1789, 1848, 1890, 1942, 2016, 2083, 2156, 2217, 2250, 2253, 2266, 2313, 2315, 2319, 2323, 2343, 2358, 2361, 2363, 2383, 2396, 2398, 2399, 2401, 2486, ...
75	59, 73, 75, 165, 181, 204, 205, 208, 283, 369, 377, 396, 427, 448, 531, 535, 573, 587, 601, 628, 664, 686, 705, 727, 836, 860, 866, 879, 907, 918, 1077, 1118, 1142, 1175, 1220, 1256, 1284, 1332, 1337, 1349, 1389, 1412, 1426, 1437, 1453, 1493, 1499, 1502, 1551, 1561, 1640, 1737, 1748, 1789, 1796, 1958, 1975, 2056, 2089, 2090, 2143, 2158, 2177, 2210,2212, 2245, 2300, 2306, 2347, 2371, 2415, 2424, 2491, ...
76	7, 29, 33, 39, 62, 69, 81, 82, 139, 165, 180, 185, 196, 253, 273, 274, 278, 337, 348, 358, 376, 389, 412, 419, 439, 507, 521, 533, 555, 571, 600, 605, 611, 621, 623, 626, 650, 658, 701, 714, 727, 733, 758, 767, 795, 817, 866, 868, 869, 878, 881, 885, 896, 905, 932, 949, 983, 1013, 1045, 1072, 1082, 1128, 1144, 1162, 1170, 1171, 1196, 1258, 1276, 1285, 1294, 1307, 1315, 1355, 1416, 1423, 1424, 1430, 1442, 1495, 1532, 1535, 1557, 1615, 1651, 1659, 1664, 1669, 1764, 1816, 1833, 1852, 1872, 1930, 1931, 1965, 2077, 2103, 2104, 2182, 2211, 2235, 2271, 2333, 2341, 2402, 2427, 2480, 2481, ...
77	2, 17, 41, 42, 90, 104, 130, 132, 186, 211, 224, 228, 235, 259, 307, 319, 362, 410, 412, 537, 603, 605, 783, 789, 795, 811, 814, 854, 858, 861, 877, 892, 906, 940, 1003, 1028, 1052, 1095, 1178, 1241, 1249, 1259, 1263, 1265, 1279, 1357, 1392, 1398, 1457, 1569, 1755, 1795, 1809, 1814, 1880, 1882, 1894, 1896, 1906, 1961, 1970, 2020, 2026, 2054, 2132,2161, 2179, 2253, 2310, 2327, 2408, 2451, ...
78	2, 8, 13, 16, 18, 41, 87, 88, 123, 134, 137, 141, 166, 180, 198, 236, 270, 288, 293, 299, 321, 361, 405, 416, 422, 450, 457, 492, 495, 514, 558, 580, 602, 605, 673, 697, 712, 713, 724, 743, 784, 805, 879, 884, 893, 1013, 1017, 1049, 1067, 1094, 1117, 1157, 1212, 1231, 1272, 1306, 1331, 1437, 1447, 1462, 1493, 1528, 1542, 1570, 1572, 1589, 1624, 1676, 1726, 1809, 1818, 1819, 1830, 1843, 1900, 1904, 1929, 1965, 1998, 2000, 2027, 2031, 2065, 2133, 2152, 2159, 2206, 2252, 2374, 2414, 2425, 2448, 2449, 2451, 2462, 2471, 2499, ...
79	22, 112, 140, 158, 159, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970, 1019, 1047, 1141, 1265, 1266, 1338, 1359, 1376, 1412, 1485, 1504, 1542, 1598, 1607, 1618, 1747, 1773, 1814, 1843, 2055, 2087, 2088, 2100, 2167, 2186, 2233, 2311, 2358, 2359, 2363, 2429, ...
80	2, 12, 13, 53, 71, 97, 123, 149, 150, 172, 197, 202, 228, 252, 286, 310, 323, 331, 335, 338, 367, 378, 400, 401, 404, 488, 504, 550, 569, 589, 615, 657, 673, 681, 699, 722, 770, 779, 787, 795, 857, 995, 1002, 1004, 1006, 1008, 1054, 1063, 1115, 1129, 1138, 1183, 1187, 1196, 1200, 1201, 1227, 1235, 1266, 1272, 1295, 1314, 1342, 1383, 1402, 1407, 1416, 1421, 1435, 1439, 1461, 1476, 1480, 1510, 1514, 1525, 1554, 1568, 1596, 1620, 1626, 1640, 1658, 1686, 1695, 1719, 1763, 1894, 1942, 2008, 2020, 2051, 2060, 2132, 2139, 2144, 2183, 2199, 2220, 2244, 2246, 2263, 2397, 2468, 2471, ...
81	21, 28, 50, 77, 79, 142, 153, 191, 194, 310, 311, 405, 440, 462, 557, 562, 638, 659, 690, 724, 746, 852, 868, 887, 919, 922, 944, 945, 1140, 1183, 1326, 1344, 1452, 1463, 1486, 1507, 1579, 1607, 1632, 1652, 1659, 1683, 1710, 1788, 1801, 1812, 1851, 1891, 1925, 1943, 1992, 2157, 2294, 2309, 2352, 2402, 2473, ...
82	61, 63, 99, 144, 230, 312, 360, 401, 413, 424, 451, 515, 542, 567, 610, 618, 622, 651, 690, 732, 817, 871, 1007, 1100, 1106, 1156, 1278, 1403, 1427, 1460, 1535, 1557, 1572, 1604, 1681, 1742, 1802, 1820, 1847, 1903, 1910, 1913, 1978, 2019, 2104, 2134, 2152, 2169, 2309, 2383, 2491, ...
83	41, 146, 386, 593, 667, 688, 906, 927, 930, 1025, 1032, 1111, 1410, 1437, 1638, 1829, 1960, 2045, 2159, 2381, 2384, ...
84	7, 12, 53, 61, 70, 71, 91, 132, 142, 156, 165, 166, 173, 187, 209, 218, 219, 236, 244, 260, 266, 270, 276, 286, 287, 306, 315, 324, 351, 361, 400, 424, 452, 461, 501, 502, 549, 569, 613, 615, 630, 631, 632, 639, 659, 669, 698, 710, 718, 721, 728, 751, 757, 782, 794, 797, 798, 826, 828, 831, 851, 853, 878, 883, 893, 895, 915, 919, 947, 965, 981, 984, 986, 1055, 1078, 1119, 1160, 1175, 1183, 1186, 1188, 1206, 1260, 1267, 1280, 1308, 1310, 1326, 1329, 1341, 1342, 1364, 1401, 1419, 1436, 1438, 1473, 1488, 1506, 1510, 1539, 1550, 1593, 1612, 1630, 1653, 1659, 1663, 1684, 1687, 1705, 1708, 1746, 1754, 1765, 1766, 1768, 1770, 1777, 1839, 1884, 1911, 1918, 1967, 1984, 1987, 1990, 2011, 2045, 2050, 2061, 2084, 2103, 2123, 2156, 2177, 2181, 2229, 2240, 2245, 2246, 2250, 2256, 2259, 2265, 2269, 2291, 2307, 2309, 2310, 2349, 2358, 2376, 2391, 2392, 2427, 2481, 2491, 2492, ...
85	2, 15, 32, 65, 77, 116, 154, 173, 197, 233, 250, 267, 392, 430, 453, 516, 558, 740, 791, 826, 844, 860, 861, 867, 868, 963, 1009, 1010, 1048, 1055, 1097, 1198, 1200, 1220, 1243, 1290, 1329, 1434, 1619, 1621, 1623, 1652, 1710, 1743, 1745, 1827, 1836, 1880, 1932, 2059, 2091, 2109, 2196, ...
86	2, 3, 6, 22, 59, 83, 91, 95, 120, 148, 195, 196, 201, 247, 252, 257, 264, 315, 360, 378, 458, 555, 680, 792, 893, 1025, 1088, 1158, 1171, 1240, 1280, 1416, 1437, 1632, 1661, 1677, 1681, 1849, 1946, 1960, 2007, 2090, 2092, 2225, 2242, 2244, 2377, 2483, ...
87	8, 27, 33, 34, 59, 118, 119, 124, 155, 189, 249, 263, 270, 275, 286, 334, 342, 382, 402, 414, 421, 468, 474, 486, 501, 530, 563, 577, 655, 697, 764, 895, 915, 932, 1108, 1175, 1188, 1228, 1234, 1268, 1277, 1308, 1357, 1358, 1370, 1373, 1400, 1491, 1519, 1529, 1558, 1560, 1658, 1693, 1699, 1735, 1745, 1773, 1805, 1866, 1876, 1905, 1915, 1961, 1996,2161, 2192, 2227, 2297, 2321, 2446, 2483, ...
88	5, 97, 131, 189, 199, 200, 258, 263, 289, 317, 352, 398, 470, 533, 554, 601, 603, 606, 732, 764, 765, 819, 869, 872, 888, 961, 1197, 1299, 1355, 1435, 1469, 1471, 1522, 1577, 1582, 1638, 1649, 1743, 1768, 1788, 1829, 1860, 1879, 1946, 1994, 2091, 2208, 2289, 2297, 2302, 2322, 2378, 2401, 2475, 2487, ...
89	2, 114, 159, 190, 234, 251, 436, 616, 834, 878, 1008, 1049, 1060, 1062, 1118, 1472, 1689, 1792, 2282, 2334, 2463, 2494, ...
90	2, 5, 6, 58, 66, 71, 78, 80, 96, 101, 122, 127, 148, 153, 183, 194, 198, 211, 244, 258, 261, 354, 387, 390, 413, 415, 446, 447, 450, 465, 475, 480, 502, 524, 532, 549, 577, 590, 607, 608, 610, 620, 623, 635, 668, 678, 716, 793, 844, 859, 862, 879, 895, 915, 958, 963, 969, 974, 1009, 1035, 1080, 1093, 1095, 1136, 1137, 1178, 1219, 1225, 1230, 1296,1301, 1309, 1315, 1352, 1359, 1369, 1375, 1381, 1383, 1389, 1411, 1450, 1453, 1463, 1514, 1599, 1638, 1651, 1682, 1692, 1705, 1710, 1734, 1768, 1803, 1850, 1883, 1936, 1965, 2037, 2041, 2066, 2108, 2111, 2114, 2117, 2120, 2145, 2164, 2195, 2210, 2225, 2287, 2293, 2327, 2382, 2387, 2389, 2406, 2407, 2458, 2470, 2472, 2483, 2485, 2498, ...
91	11, 19, 28, 30, 46, 80, 116, 140, 159, 198, 272, 331, 357, 360, 448, 544, 728, 827, 854, 910, 937, 1019, 1041, 1106, 1164, 1174, 1226, 1229, 1287, 1307, 1327, 1348, 1408, 1471, 1473, 1618, 1642, 1751, 1810, 1900, 1965, 1976, 2051, 2110, 2206, 2226, 2254, 2331, ...
92	4, 5, 12, 19, 43, 47, 70, 164, 183, 225, 258, 261, 287, 297, 306, 323, 335, 340, 351, 363, 383, 397, 442, 534, 545, 558, 572, 594, 624, 652, 656, 659, 666, 673, 681, 696, 701, 724, 730, 773, 810, 887, 923, 933, 972, 997, 1002, 1006, 1061, 1124, 1135, 1178, 1196, 1225, 1230, 1313, 1356, 1407, 1409, 1451, 1452, 1483, 1576, 1586, 1602, 1607, 1612, 1619, 1667, 1691, 1697, 1823, 1827, 1935, 1963, 1992, 2035, 2099, 2138, 2156, 2199, 2252, 2274, 2299, 2393, 2398, 2447, 2477, ...
93	2, 10, 15, 22, 27, 63, 98, 113, 142, 156, 188, 236, 278, 282, 293, 300, 315, 333, 353, 374, 420, 487, 509, 558, 580, 592, 793, 924, 1099, 1131, 1195, 1242, 1268, 1274, 1321, 1357, 1413, 1484, 1542, 1547, 1611, 1683, 1699, 1702, 1711, 1730, 1746, 1760, 1819, 1854, 1895, 1902, 2026, 2077, 2092, 2111, 2151, 2222, 2272, 2328, 2388, 2390, 2409, 2450, 2475, 2482, 2483, ...
94	6, 7, 17, 90, 126, 139, 143, 257, 293, 295, 319, 328, 387, 482, 519, 603, 720, 751, 819, 884, 896, 903, 905, 921, 952, 954, 956, 1042, 1058, 1147, 1170, 1237, 1253, 1279, 1295, 1343, 1366, 1370, 1406, 1465, 1514, 1593, 1595, 1607, 1609, 1622, 1701, 1705, 1709, 1736, 1772, 1791, 1833, 1860, 1866, 1879, 1969, 1990, 1992, 2059, 2115, 2132, 2169, 2219,2233, 2253, 2344, ...
95	44, 47, 66, 155, 187, 344, 345, 448, 492, 525, 546, 611, 642, 704, 721, 748, 820, 858, 1062, 1108, 1112, 1195, 1277, 1314, 1453, 1558, 1597, 1667, 1725, 1760, 1780, 1793, 1807, 1901, 1963, 1983, 2021, 2055, 2060, 2251, 2266, 2331, ...
96	4, 19, 28, 36, 44, 79, 94, 106, 109, 125, 132, 147, 319, 343, 399, 412, 415, 435, 441, 449, 463, 491, 562, 576, 625, 643, 646, 649, 691, 701, 712, 787, 788, 820, 824, 840, 851, 976, 994, 1047, 1079, 1111, 1112, 1136, 1195, 1216, 1219, 1225, 1291, 1309, 1339, 1359, 1425, 1456, 1466, 1520, 1525, 1533, 1540, 1543, 1601, 1633, 1668, 1679, 1738, 1744, 1746, 1789, 1854, 1873, 1922, 2056, 2101, 2110, 2159, 2225, 2266, 2267, 2286, 2301, 2320, 2346, 2378, 2395, 2424, 2468, 2488, ...
97	12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934, 1054, 1114, 1116, 1128, 1144, 1159, 1193, 1364, 1788, 2086, 2215, 2254, 2448, 2461, ...
98	2, 19, 20, 40, 52, 75, 80, 88, 95, 98, 100, 103, 106, 123, 124, 162, 176, 198, 220, 276, 311, 358, 396, 427, 429, 522, 525, 529, 555, 576, 586, 605, 675, 695, 733, 782, 811, 833, 872, 879, 891, 898, 950, 952, 971, 977, 982, 1076, 1090, 1150, 1156, 1219, 1231, 1257, 1260, 1312, 1340, 1483, 1621, 1707, 1716, 1742, 1774, 1806, 1810, 1847, 1850, 1926,1935, 1975, 2019, 2042, 2090, 2153, 2167, 2169, 2238, 2260, 2262, 2306, 2346, 2356, 2451, 2474, ...
99	63, 88, 113, 123, 134, 146, 217, 284, 320, 352, 504, 511, 532, 544, 603, 704, 802, 840, 870, 974, 1205, 1242, 1278, 1414, 1441, 1471, 1516, 1579, 1660, 1740, 1763, 1765, 1846, 1874, 1889, 1932, 1939, 1980, 2095, 2132, 2136, 2396, 2493, ...
100	20, 22, 88, 106, 114, 133, 179, 189, 222, 224, 234, 312, 326, 333, 355, 440, 443, 492, 509, 535, 615, 676, 727, 751, 765, 767, 812, 813, 845, 925, 988, 1032, 1089, 1106, 1229, 1249, 1259, 1281, 1323, 1343, 1434, 1435, 1461, 1482, 1536, 1621, 1649, 1678, 1692, 1739, 1779, 1824, 1979, 2020, 2021, 2033, 2052, 2164, 2186, 2322, 2391, 2418, 2438, 2457,...

prime p	p가 유일무이한 전성기라는 근거.
2	3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, ...
3	2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887, 4373, 4607, 5183, 5831, 6143, 6560, 6562, 6911, 7775, 8747, 9215, 10367, 11663, 12287, 13121, 13823, 15551,17495, 18431, 19682, 19684, 20735, 23327, 24575, 26243, 27647, 31103, 34991, 36863, 39365, 41471, 46655, 49151, 52487, 55295, 59048, 59050, 62207, 69983, 73727, 78731, 82943, 93311, 98303, 104975, 110591, 118097, 124415, 139967, 147455, 157463, 165887, 177146, 177148, 186623, 196607, 209951, 221183, 236195, 248831, 279935, 294911, 314927, 331775,354293, 373247, 393215, 419903, 442367, 472391, 497663, 531440, 531442, 559871, 589823, 629855, 663551, 708587, 746495, 786431, 839807, 884735, 944783, 995327, ...
5	2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479, 24999, 25599, 31249, 31999, 39999, 40959, 49999, 51199, 62499, 63999, 78124, 78126, 79999, 81919, 99999, 102399, 124999, 127999, 156249, 159999, 163839, 199999, 204799, 249999, 255999, 312499, 319999, 327679, 390624, 390626, 399999, 409599, 499999, 511999, 624999, 639999, 655359, 781249, 799999, 819199, 999999, ...
7	2, 3, 4, 5, 6, 8, 13, 18, 19, 27, 48, 50, 55, 97, 111, 195, 223, 342, 344, 391, 447, 685, 783, 895, 1371, 1567, 1791, 2400, 2402, 2743, 3135, 3583, 4801, 5487, 6271, 7167, 9603, 10975, 12543, 14335, 16806, 16808, 19207, 21951, 25087, 28671, 33613, 38415, 43903, 50175, 57343, 67227, 76831, 87807, 100351, 114687, 117648, 117650, 134455, 153663, 175615, 200703, 229375, 235297, 268911, 307327, 351231, 401407, 458751, 470595, 537823, 614655, 702463, 802815, 823542, 823544, 917503, 941191, ...
11	2, 3, 10, 12, 21, 43, 87, 120, 122, 175, 241, 351, 483, 703, 967, 1330, 1332, 1407, 1935, 2661, 2815, 3871, 5323, 5631, 7743, 10647, 11263, 14640, 14642, 15487, 21295, 22527, 29281, 30975, 42591, 45055, 58563, 61951, 85183, 90111, 117127, 123903, 161050, 161052, 170367, 180223, 234255, 247807, 322101, 340735, 360447, 468511, 495615, 644203, 681471,720895, 937023, 991231, ...
13	2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111, 371292, 371294, 425983, 456975, 562431, 692223, 742585, 851967, 913951, ...
17	2, 4, 16, 18, 33, 67, 135, 271, 288, 290, 543, 577, 1087, 1155, 2175, 2311, 4351, 4623, 4912, 4914, 8703, 9247, 9825, 17407, 18495, 19651, 34815, 36991, 39303, 69631, 73983, 78607, 83520, 83522, 139263, 147967, 157215, 167041, 278527, 295935, 314431, 334083, 557055, 591871, 628863, 668167, ...
19	2, 7, 8, 18, 20, 37, 75, 151, 303, 360, 362, 607, 721, 1215, 1443, 2431, 2887, 4863, 5775, 6858, 6860, 9727, 11551, 13717, 19455, 23103, 27435, 38911, 46207, 54871, 77823, 92415, 109743, 130320, 130322, 155647, 184831, 219487, 260641, 311295, 369663, 438975, 521283, 622591, 739327, 877951, ...
23	22, 24, 45, 91, 183, 367, 528, 530, 735, 1057, 1471, 2115, 2943, 4231, 5887, 8463, 11775, 12166, 12168, 16927, 23551, 24333, 33855, 47103, 48667, 67711, 94207, 97335, 135423, 188415, 194671, 270847, 279840, 279842, 376831, 389343, 541695, 559681, 753663, 778687, ...
29	28, 30, 41, 57, 115, 231, 463, 840, 842, 927, 1681, 1855, 3363, 3711, 6727, 7423, 13455, 14847, 24388, 24390, 26911, 29695, 48777, 53823, 59391, 97555, 107647, 118783, 195111, 215295, 237567, 390223, 430591, 475135, 707280, 707282, 780447, 861183, 950271, ...
31	2, 5, 6, 30, 32, 61, 123, 247, 495, 960, 962, 991, 1921, 1983, 3843, 3967, 7687, 7935, 15375, 15871, 29790, 29792, 30751, 31743, 59581, 61503, 63487, 119163, 123007, 126975, 238327, 246015, 253951, 476655, 492031, 507903, 923520, 923522, 953311, 984063, ...
37	6, 10, 11, 36, 38, 73, 147, 295, 591, 1183, 1368, 1370, 2367, 2737, 4735, 5475, 9471, 10951, 18943, 21903, 37887, 43807, 50652, 50654, 75775, 87615, 101305, 151551, 175231, 202611, 303103, 350463, 405223, 606207, 700927, 810447, ...
41	2, 3, 4, 9, 40, 42, 81, 163, 327, 655, 1311, 1680, 1682, 2623, 3361, 5247, 6723, 10495, 13447, 20991, 26895, 41983, 53791, 68920, 68922, 83967, 107583, 137841, 167935, 215167, 275683, 335871, 430335, 551367, 671743, 860671, ...
43	2, 6, 7, 42, 44, 85, 171, 343, 687, 1375, 1848, 1850, 2751, 3697, 5503, 7395, 11007, 14791, 22015, 29583, 44031, 59167, 79506, 79508, 88063, 118335, 159013, 176127, 236671, 318027, 352255, 473343, 636055, 704511, 946687, ...
47	46, 48, 93, 187, 375, 751, 1503, 2208, 2210, 3007, 4417, 6015, 8835, 12031, 17671, 24063, 35343, 48127, 70687, 96255, 103822, 103824, 141375, 192511, 207645, 282751, 385023, 415291, 565503, 770047, 830583, ...
53	52, 54, 105, 211, 423, 847, 1695, 2808, 2810, 3391, 5617, 6783, 11235, 13567, 22471, 27135, 44943, 54271, 89887, 108543, 148876, 148878, 179775, 217087, 297753, 359551, 434175, 595507, 719103, 868351, ...
59	58, 60, 117, 235, 471, 943, 1887, 3480, 3482, 3775, 6961, 7551, 13923, 15103, 27847, 30207, 55695, 60415, 111391, 120831, 205378, 205380, 222783, 241663, 410757, 445567, 483327, 821515, 891135, 966655, ...
61	3, 11, 13, 14, 60, 62, 121, 243, 487, 975, 1951, 3720, 3722, 3903, 7441, 7807, 14883, 15615, 29767, 31231, 59535, 62463, 119071, 124927, 226980, 226982, 238143, 249855, 453961, 476287, 499711, 907923, 952575, 999423, ...
67	66, 68, 133, 267, 535, 1071, 2143, 4287, 4488, 4490, 8575, 8977, 17151, 17955, 34303, 35911, 68607, 71823, 137215, 143647, 274431, 287295, 300762, 300764, 548863, 574591, 601525, ...
71	70, 72, 141, 283, 567, 1135, 2271, 4543, 5040, 5042, 9087, 10081, 18175, 20163, 36351, 40327, 72703, 80655, 145407, 161311, 290815, 322623, 357910, 357912, 581631, 645247, 715821, ...
73	2, 3, 8, 9, 72, 74, 145, 291, 583, 1167, 2335, 4671, 5328, 5330, 9343, 10657, 18687, 21315, 37375, 42631, 74751, 85263, 149503, 170527, 299007, 341055, 389016, 389018, 598015, 682111, 778033, ...
79	78, 80, 157, 315, 631, 1263, 2527, 5055, 6240, 6242, 10111, 12481, 20223, 24963, 40447, 49927, 80895, 99855, 161791, 199711, 323583, 399423, 493038, 493040, 647167, 798847, 986077, ...
83	82, 84, 165, 331, 663, 1327, 2655, 5311, 6888, 6890, 10623, 13777, 21247, 27555, 42495, 55111, 84991, 110223, 169983, 220447, 339967, 440895, 571786, 571788, 679935, 881791, ...
89	88, 90, 177, 355, 711, 1423, 2847, 5695, 7920, 7922, 11391, 15841, 22783, 31683, 45567, 63367, 91135, 126735, 182271, 253471, 364543, 506943, 704968, 704970, 729087, ...
97	96, 98, 193, 387, 775, 1551, 3103, 6207, 9408, 9410, 12415, 18817, 24831, 37635, 49663, 75271, 99327, 150543, 198655, 301087, 397311, 602175, 794623, 912672, 912674, ...

참고 문헌 목록

크리스 K.Caldwell, Harvey Dubner, "Unique period primes", Journal of Rediation Mathics 29:1:43-48 (1998) 사전 인쇄

참조

^ Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.
^ PRP 레코드: 10000 상위 확률
^ The Top Tween Unique; Chris Caldwell
^ PRP 레코드
^ 커닝햄 프로젝트
^ PRP 레코드

Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.

외부 링크

[1] Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.

[2] PRP 레코드: 10000 상위 확률

[3] The Top Tween Unique; Chris Caldwell

[4] PRP 레코드

[5] 커닝햄 프로젝트

[6] PRP 레코드

[2]

[3]

[4]

[5]

v t 프라임 번호 클래스
수식별	페르마( $2 2 n$ + $1$ ) 메르센( $2 p$ - $1$ ) 더블 메르센( $2 2 p -1$ - $1$ ) 와그스태프( $2 p +1)/3$ 프로트( $k /2 n$ + $1$ ) 요인( $n!$ ± $1$ ) 원시적( $p n #$ ± $1$ ) 유클리드( $p n #$ + $1$ ) 피타고라스 ( $4n$ + 1) 삐에폰트(2 $/3 m n$ + $1$ ) 쿼탄( $x 4 4$ + y) 솔리나스( $2 m$ ± 2 ± $1 n$ ) 컬렌 ( $n \cdot2 n$ + $1$ ) 우달( $n /2 n$ - $1$ ) 쿠바어( $x 3 3$ - y $)/(x$ - $y$ ) 레이랜드( $x y x$ + y) 타비트( $3/2 n$ - $1$ ) 윌리엄스 ( $b-1)\cdot b n$ - $1$ ) 밀스(⌊ $A 3 n ⌋)$
정수순별	피보나치 루카스 펠 뉴먼-상크스-윌리엄스 페린 파티션 벨 모츠킨
재산별	위페리히 (페어) 월선 월스텐홀름 윌슨 럭키 행운의 라마누잔 필라이 정규 강하다 선미 초점수(엘리프틱 곡선) 슈퍼싱글러 (달빛설) 좋아 잘 하는 군요 힉스 매우 편협한
기저 의존적	팔린드로믹 에미르프 재단위 $(10 n$ - 1 $)/9$ 퍼머블 원형 잘릴 수 있음 미니멀 섬세한 프라임벌 풀 파충류 유니크 행복하다 셀프 스마란다체-웰린 스트로보그램법 디헤드랄 테트라디어
패턴	트윈( $p,$ p + $2$ ) 바이-트윈 체인( $n$ - 1 $, n$ + 1 $, 2n$ - 1 $, 2n$ + 1, $\dots)$ 트리플릿( $p,$ $p$ + 2 $또는$ $p$ + 4 $, p$ + $6$ ) 쿼드러플릿( $p,$ $p$ + 2 $,$ $p$ + 6 $, p$ + $8$ ) k-투플 사촌( $p,$ p + $4$ ) 섹시( $p,$ p + $6$ ) 첸 소피 제르맹/세이프( $p, 2p$ + $1$ ) 커닝햄( $p, 2p$ ± 1, $4p$ ± 3, $8p$ ± 7, $...)$ 산술수열( $p$ + $a\cdotn,$ $n$ = $0,$ 1, 2 $, 3$ , $...)$ 균형( $연속$ $p$ - $n,$ $p,$ p + $n$ )
크기별	메가(100,000자리 이상) 알려진 것 중 가장 큰 것
콤플렉스	아이젠슈타인 전성기 가우스 프라임
합성수	가성비 카탈루냐 주 타원체 오일러 오일러자코비 페르마 프로베니우스 루카스 소머루카스 강하다 카마이클 수 거의 전성기 세미프라임 인터프라임 치명적
관련 항목	개연성 프라임 산업용 프라임 불법 프라임 소수점 수식 프라임 갭
처음 60회	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
소수 목록

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