This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A112633 #29 Oct 19 2024 08:34:58
%S A112633 3,7,19,31,107,127,607,1279,2203,4423,86243,110503,216091,756839,
%T A112633 1257787,20996011,24036583,25964951,37156667
%N A112633 Mersenne prime indices that are also Gaussian primes.
%C A112633 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!.
%F A112633 The intersection of A000043 and A002145. - _R. J. Mathar_, Oct 06 2008
%t A112633 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 *)
%t A112633 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 *)
%t A112633 Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 120] == 8 &] (* _Amiram Eldar_, Oct 19 2024 *)
%o A112633 (Python)
%o A112633 from itertools import count, islice
%o A112633 from sympy import isprime, prime
%o A112633 def A112633_gen(): # generator of terms
%o A112633 return filter(lambda p: p&2 and isprime((1<<p)-1), (prime(n) for n in count(2)))
%o A112633 A112633_list = list(islice(A112633_gen(),10)) # _Chai Wah Wu_, Mar 21 2023
%Y A112633 Cf. A112634, A112648, A112649.
%Y A112633 Cf. also A000043, A000668, A002145, A124477, A139484, A145039, A145040, A145041, A145042.
%K A112633 nonn,more
%O A112633 1,1
%A A112633 _Jorge Coveiro_, Dec 27 2005
%E A112633 Edited by _N. J. A. Sloane_, Jan 06 2018
%E A112633 a(19) from _Ivan Panchenko_, Apr 12 2018