A112633 Mersenne prime indices that are also Gaussian primes.

3, 7, 19, 31, 107, 127, 607, 1279, 2203, 4423, 86243, 110503, 216091, 756839, 1257787, 20996011, 24036583, 25964951, 37156667

Offset: 1

Author

Jorge Coveiro, Dec 27 2005

Keywords

• nonn
• more

Comments

Also, primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 7 mod 5!. - Artur Jasinski, Sep 30 2008. Proof that this is the same sequence, from Jeppe Stig Nielsen, Jan 02 2018: An odd index p>2 will be either 1 or 3 mod 4. If it is 1, then 2^p = 2^(4k+1) will be 2 mod 5, and be 0 mod 4, and be 2 mod 3. This completely determines 2^p (and hence 2^p - 1) mod 5!. The other case, when p is 3 mod 4, will make 2^p congruent to 3 mod 5, to 0 mod 4, and to 2 mod 3. This leads to the other (distinct) value of 2^p mod 5!.

Crossrefs

Cf. A112634, A112648, A112649.

Cf. also A000043, A000668, A002145, A124477, A139484, A145039, A145040, A145041, A145042.

Programs

Mathematica

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 5! ] == 7, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (* Artur Jasinski, Sep 30 2008 *)
Select[{2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,4423, 9689,9941,11213,19937,21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269,2976221,3021377,6972593, 13466917,20996011, 24036583,25964951,30402457,32582657,37156667,43112609}, Mod[2^#-1,120]==7&] (* Harvey P. Dale, Nov 26 2013 *)
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 120] == 8 &] (* Amiram Eldar, Oct 19 2024 *)

Python

from itertools import count, islice
from sympy import isprime, prime
def A112633_gen(): # generator of terms
    return filter(lambda p: p&2 and isprime((1<A112633_list = list(islice(A112633_gen(),10)) # Chai Wah Wu, Mar 21 2023

Formula

The intersection of A000043 and A002145. - R. J. Mathar, Oct 06 2008

Extensions

Edited by N. J. A. Sloane, Jan 06 2018

a(19) from Ivan Panchenko, Apr 12 2018