A125854 Primes p with the property that p divides the Wolstenholme number A001008((p+1)/2).

3, 29, 37, 3373, 2001907169

Offset: 1

Author

Alexander Adamchuk, Dec 11 2006

Keywords

• hard
• more
• nonn

Comments

Note that if prime p>3 divides A001008((p+1)/2) then it also divides A001008((p-3)/2).

Note that for a prime p, H([p/2]) == 2*(2^(-p(p-1)) - 1)/p^2 (mod p). Therefore a prime p divides the Wolstenholme number A001008((p+1)/2) if and only if 2^(-p(p-1)) == 1 - p^2 (mod p^3) or, equivalently, 2^(p-1) == 1 + p (mod p^2).

Disjunctive union of the sequences A154998 and A121999 that contain primes congruent respectively to 1,3 and 5,7 modulo 8. (Alekseyev)

a(6) > 5.5*10^12. - Giovanni Resta, Apr 13 2017

Primes p that are base-((p-1)/2) Wieferich primes, that is, primes p such that ((p-1)/2)^(p-1) == 1 (mod p^2). - Jianing Song, Jan 27 2019

Examples

			a(1) = 3 because prime 3 divides A001008(2) = 3 and there is no p < 3 that divides A001008((p+1)/2).
a(2) = 29 because 29 divides A001008(15) = 1195757 and there is no prime p (3 < p < 29) that divides A001008((p+1)/2).
		

Crossrefs

Cf. A001008, A121999, A014566, A154998.

Programs

Mathematica

Select[Prime[Range[1, 5000]],
Divisible[Numerator[HarmonicNumber[(# + 1)/2]], #] &] (* Robert Price, May 10 2019 *)

Extensions

Entry revised and a(5) = 2001907169 provided by Max Alekseyev, Jan 18 2009

Edited by Max Alekseyev, Oct 13 2009