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 A121371 #9 Jul 23 2021 05:30:45
%S A121371 1,3,5,8,99,275,278,404,96,1538,1253,15858,189168,119552,221444,
%T A121371 1047122,3571449,5424924,1575995
%N A121371 Least number k such that (k*M(n))^2 + k*M(n) - 1 and (k*M(n))^2 + k*M(n) + 1 are twin primes where M(n) is the n-th Mersenne prime.
%e A121371 M(2) = 2^3-1 = 7, 7^2+7-1 = 55 is composite, (2*7)^2+2*7-1 = 209 is composite,
%e A121371 (3*7)^2+3*7-1 = 461 is prime, 461 and 463 are twin primes, so a(2) = 3.
%t A121371 f[p_] := Module[{k = 1}, While[!PrimeQ[(k*p)^2 + k*p - 1] || !PrimeQ[(k*p)^2 + k*p + 1], k++]; k]; f /@ (2^MersennePrimeExponent[Range[10]] - 1) (* _Amiram Eldar_, Jul 23 2021 *)
%Y A121371 Cf. A000043, A000668, A001097, A121370.
%K A121371 nonn,hard,more
%O A121371 1,2
%A A121371 _Pierre CAMI_, Jul 24 2006
%E A121371 a(8) corrected and a(18)-a(19) added by _Amiram Eldar_, Jul 23 2021