A129828 Odd primes p such that p divides both Apery numbers A005259((p-3)/2) and A005259((p+1)/2).

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

Offset: 1

Author

Alexander Adamchuk, May 20 2007

Keywords

• nonn
• hard
• more

Comments

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

Links

• Eric Weisstein's World of Mathematics, Apéry Number.

Crossrefs

Cf. A005259 (Apéry numbers).

Programs

Mathematica

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} ]

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

Extensions

a(7) from Alexander Adamchuk, May 02 2010