A129828 - OEIS

%I #17 Oct 28 2023 00:25:02

%S 5,19,37,283,367,3347,132157

%N Odd primes p such that p divides both Apery numbers A005259((p-3)/2) and A005259((p+1)/2).

%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AperyNumber.html">Apéry Number</a>.

%t 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,2,10000} ].

%o (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 Cf. A005259 (Apéry numbers).

%K nonn,hard,more

%O 1,1

%A _Alexander Adamchuk_, May 20 2007

%E a(7) from _Alexander Adamchuk_, May 02 2010