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 A129828 #21 Aug 06 2025 01:03:05
%S A129828 5,19,37,283,367,3347,132157
%N A129828 Odd primes p such that p divides both Apery numbers A005259((p-3)/2) and A005259((p+1)/2).
%C A129828 From the recurrence for A005259, it follows that A005259((p-3)/2) == A005259((p+1)/2) (mod p) for any odd prime p. Hence, p divides either both or neither of A005259((p-3)/2) and A005259((p+1)/2). There are no other terms below 10^6. - _Max Alekseyev_, Oct 27 2023
%H A129828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AperyNumber.html">Apéry Number</a>.
%t A129828 a=5; b=73; Do[ c=a; a=b; b=( (34*n^3 + 51*n^2 + 27*n +5)*b - n^3*c )/(n+1)^3; If[ PrimeQ[2n-1] && IntegerQ[a/(2n-1)], Print[2n-1] ], {n,2,70000} ]
%o A129828 (PARI) { is_A129828(p) = my(n,n1,n2,f,s=1); n1=n2=f=Mod(1,p); n=(p-3)\2; for(k=1,n, f*=k; n1*=n+k; n2*=n+1-k; s+=(n1*n2/f^2)^2); s==0; } \\ _Max Alekseyev_, Oct 27 2023
%Y A129828 Cf. A005259 (Apéry numbers).
%K A129828 nonn,hard,more
%O A129828 1,1
%A A129828 _Alexander Adamchuk_, May 20 2007
%E A129828 a(7) from _Alexander Adamchuk_, May 02 2010