A177783 Wolstenholme quotient of prime p=A000040(n), i.e., such integer m

3, 6, 6, 7, 10, 14, 18, 20, 16, 24, 17, 38, 39, 19, 29, 28, 12, 53, 31, 19, 53, 58, 48, 42, 1, 33, 53, 37, 5, 81, 4, 17, 29, 13, 13, 72, 75, 70, 173, 159, 111, 150, 39, 178, 106, 163, 196, 163, 172, 30, 98, 24, 177, 261, 212, 223, 122, 147, 276, 17, 92, 111, 27, 209, 241

Offset: 3

Author

Max Alekseyev, May 13 2010

Keywords

• nonn

Comments

a(n) = 0 iff A000040(n) is a Wolstenholme prime (given by A088164).

For n>2 and p=A000040(n), H(p^2-p) == H(p^2-1) == a(n)*p (mod p^2).

Links

• David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
• R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
• Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, Journal of Number Theory, Volume 123, Issue 1, March 2007, Pages 18-26.

Crossrefs

Cf. A034602, A072984, A087754, A092101, A092103, A092193, A128673.

Programs

PARI

{ a(n) = my(p); p=prime(n); ((binomial(2*p-1,p)-1)/2/p^3)%p }

Formula

a(n) = H(p-1)/p^2 mod p = A001008(p-1)/A002805(p-1)/p^2 mod p = A034602(n)/2 mod p = (binomial(2*p-1,p)-1)/(2*p^3) mod p, where p = A000040(n).

a(n) = (-1/3)*B(p-3) mod p, with p=prime(n) and B(n) is the n-th Bernoulli number. - Michel Marcus, Feb 05 2016

a(n) = A087754(n)/4 mod A000040(n).

Extensions

Edited by Max Alekseyev, May 16 2010