A244919 For odd prime p, largest k such that binomial(2p-1, p-1) is congruent to 1 modulo p^k.

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset

2,1

Comments

Wolstenholme's theorem implies that k >= 3 for all p > 3. The prime p is a Wolstenholme prime if and only if k > 3. For the primes up to 10^9 this holds only for p = 16843 and p = 2124679, where in each case a(n) = 4 (i.e. a(1944) = 4 and a(157504) = 4).

Links

Table of n, a(n) for n=2..73.

R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arith., Volume 71, Issue 4 (1995), 381-389.

R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

Prog

(PARI) forprime(p=3, 10^3, k=1; while(Mod(binomial(2*p-1, p-1), p^k)==1, j=k; k++); if(Mod(binomial(2*p-1, p-1), p^k)!=1, print1(j, ", ")))

Crossrefs

Cf. A034602, A088164.

Sequence in context: A340944 A270533 A344511 * A158799 A157532 A065684

Adjacent sequences: A244916 A244917 A244918 * A244920 A244921 A244922

Keyword

nonn

Author

Felix Fröhlich, Jul 08 2014

Status

approved