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 A308404 #27 Aug 04 2019 06:38:57
%S A308404 1,3,57,1623,2388747,4989275679,640689916425033,1822252163947383837,
%T A308404 974834644028245238101857699,55649241817444349958527998041,
%U A308404 36596034629737014817675324057147576383,126872100333877939558050221738699065414707
%N A308404 a(n) = Q(A001359(n)), with Q(n) = (2^(n+2)-3n-8)/(n*(n+2)).
%C A308404 Aebi and Cairns proved that if (p, p+2) are twin primes, then 2^(p+2) == 3p+8 (mod p(p+2)). This sequence contains the quotients of this congruence.
%H A308404 Christian Aebi and Grant Cairns, <a href="http://doi.org/10.4171/EM/103">Catalan numbers, primes, and twin primes</a>, Elemente der Mathematik, Vol. 63, No. 4 (2008), pp. 153-164.
%e A308404 a(2) = 3, because A001359(2) = 5 and Q(5) = (2^(5+2)-3*5-8)/(5*(5+2)) = 3.
%t A308404 s={}; Do[If[PrimeQ[n] && PrimeQ[n+2], q = (2^(n+2)-3n-8)/(n(n+2)); AppendTo[s,q]], {n, 1, 1000}]; s
%o A308404 (PARI) lista(nn) = {forprime(p=2, nn, if (isprime(p+2), print1((2^(p+2)-3*p-8)/(p*(p+2)), ", ");););} \\ _Michel Marcus_, Aug 04 2019
%Y A308404 Cf. A001359, A224695, A292691.
%K A308404 nonn
%O A308404 1,2
%A A308404 _Amiram Eldar_, Aug 04 2019