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 A139424 #15 May 09 2019 19:03:30
%S A139424 1,1,1,1,1,43,1,41,53,91,317,317,43,1,37,3595,563,17239,911,11497,
%T A139424 58501,1259,10283,138569,72247,27733,11777,179105
%N A139424 Smallest number k such that M(n)^2-k*M(n)-1 is prime with M(n) = Mersenne primes = A000668(n).
%C A139424 All primes certified using openpfgw_v12 from primeform group
%e A139424 3*3-1*3-1=5 prime 3=M(1)=2^2-1 so k(1)=1;
%e A139424 7*7-1*7-1=41 prime 7=M(2)=2^3-1 so k(2)=1;
%e A139424 31*31-1*31-1=929 prime 31=M(3)=2^5-1 so k(3)=1.
%t A139424 A000043 = {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, 42643801, 43112609};
%t A139424 Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
%t A139424 While[! PrimeQ[m2 - k*m - 1], k++]; k, {n, 15}] (* _Robert Price_, Apr 17 2019 *)
%Y A139424 Cf. A000668, A139425, A139426, A139427, A139428, A139429, A139430, A139421.
%K A139424 hard,more,nonn
%O A139424 1,6
%A A139424 _Pierre CAMI_, Apr 21 2008
%E A139424 a(27)-a(28) from _Robert Price_, May 09 2019