OFFSET
1,1
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!.
FORMULA
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 *)
PROG
(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<<p)-1), (prime(n) for n in count(2)))
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
Jorge Coveiro, Dec 27 2005
EXTENSIONS
Edited by N. J. A. Sloane, Jan 06 2018
a(19) from Ivan Panchenko, Apr 12 2018
STATUS
approved