A188133 Primes p such that 10p+1 divides 2^p-1.

43, 487, 547, 571, 883, 1459, 1663, 1723, 2539, 3319, 3511, 4903, 5107, 5431, 6199, 6367, 8011, 8599, 9007, 9391, 9511, 10111, 11119, 11959, 12379, 12703, 13291, 13339, 13999, 14083, 14551, 14767, 15187, 15319, 15859, 15991, 16183, 16603, 16747, 17659, 18427, 19699

Offset: 1

Author

M. F. Hasler, Mar 21 2011

Keywords

• nonn

Comments

It is known that divisors of M(p)=2^p-1 are of the form 2kp+1. For k=1, these are the Lucasian primes A002515, for k=2 there are no such divisors, for k=3 these divisors are listed in A188130 and for k=4 in A122095.

The equivalent sequence of prime indices is 14, 93, 101, 105, 153, 232, 261, 269, ....

If k == 2 (mod 4), there are no such divisors in general and here there are no smaller k's than k = 5. - Karl-Heinz Hofmann, Jan 27 2022

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A350702 (k = 7).

Cf. A001348, A000040.

Programs

Mathematica

Select[Range[2*10^4], PrimeQ[#] && PowerMod[2, #, 10# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)
Select[Prime[Range[2500]],PowerMod[2,#,10#+1]==1&] (* Harvey P. Dale, Dec 08 2024 *)

PARI

forprime(p=1,1e5, Mod(2,p*10+1)^p-1 || print1(p", "))

Python

from sympy import sieve
print([p for p in sieve[1:10000] if pow(2,p,10*p+1) == 1])
# Karl-Heinz Hofmann, Jan 27 2022

Formula

{p = A000040(i): 10*p+1 | A001348(i)}. - R. J. Mathar, Mar 21 2011