A350702 Primes p such that 14*p + 1 divides 2^p - 1.

929, 1433, 2393, 2609, 2657, 4373, 4793, 6029, 7529, 10133, 10433, 10949, 10973, 13049, 13109, 16829, 18869, 20873, 22349, 23417, 24137, 26717, 27737, 27893, 28433, 28517, 30977, 33809, 33857, 37217, 38189, 38237, 39209, 39749, 41453, 41813, 42569, 43313, 43613

Offset: 1

Author

Karl-Heinz Hofmann, Jan 27 2022

Keywords

• nonn

Comments

Known divisors of Mersenne(p) (2^p-1 or Mp for short) are of the form 2*k*p+1. See crossrefs for other k's. If k == 2 (mod 4), there are no such divisors in general. Here k is 14/2 = 7.

Examples

			See LINKS for example of a(13).
		

Links

• Karl-Heinz Hofmann, Mersenne(p) table with k = 7 and, if they exist, additional k < 7 plus their corresponding factors.
• mersenne.ca, a(13) = M10973 Mersenne number exponent details.
• mersenne.ca, a(1..995) at GIMPS with k = 7 and Exponent 3..2000000.

Crossrefs

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A188133 (k = 5).

Cf. A000040, A001348.

Programs

Mathematica

Select[Range[45000], PrimeQ[#] && PowerMod[2, #, 14*# + 1] == 1 &] (* Amiram Eldar, Jan 27 2022 *)

PARI

forprime(p=1, 1e6, if (Mod(2, p*14+1)^p==1, print1(p,", ")))

Python

from sympy import sieve
print([p for p in sieve[1:1000000] if pow(2, p, 14*p+1) == 1])

Formula

{p = A000040(i): 14*p+1 | A001348(i)}.