A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

1, 5, 7, 71, 379, 2659

Offset: 1

Author

Alexander Adamchuk, Mar 10 2007

Keywords

• hard
• more
• nonn

Comments

For k > 1 all 5 listed terms are primes.

The only known numbers k such that k divides the numerator of alternating Harmonic number H'((k-1)/2) = A058313((k-1)/2) are the Wieferich primes (A001220): 1093 and 3511.

An odd prime p = prime(n) belongs to this sequence iff Fermat quotient A007663(n) == A130912(n) == 2*(-1)^((p+1)/2) (mod p). - Max Alekseyev, Nov 30 2022

Links

• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A001008, A007663, A001220, A058313, A125854, A121999, A130912.

Programs

Mathematica

f=0; Do[ f = f + (-1)^(n+1)*1/n; g = Numerator[f]; If[ IntegerQ[ g/(2n-1) ], Print[2n-1]], {n,1,3000} ]

Extensions

Edited by Max Alekseyev, Nov 30 2022