A244214 a(n) = binomial(2*c-1, c-1) (mod c^3), where c is the n-th composite.

35, 30, 291, 253, 378, 782, 2404, 1260, 291, 3378, 410, 7899, 3996, 6030, 126, 10988, 11188, 5180, 19712, 8483, 5334, 34394, 1841, 21410, 20580, 39556, 38810, 64260, 35972, 66060, 36504, 61326, 1716, 123628, 49140, 63748, 124392, 20091, 99388, 157767, 24392

Offset: 1

Author

Felix Fröhlich, Jun 23 2014

Keywords

• nonn

Comments

A counterexample to the converse of Wolstenholme's theorem (CWT) must have a(n) = 1. No such counterexample is known and if CWT holds, then a(n) > 1 for all n. If the square of a prime p is a counterexample to CWT, that prime satisfies the Wolstenholme congruence modulo p^6 (Cf. McIntosh (1995), p. 387).

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..10000
• C. Helou and G. Terjanian, On Wolstenholme's theorem and its converse, Journal of Number Theory, Volume 128, Issue 3 (2008), 475-499.
• R. J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.
• V. Trevisan and K. Weber, Testing the converse of Wolstenholme's theorem, Matematica Contemporanea, 21 (2001), 275-286.
• Wikipedia, Wolstenholme's theorem

Crossrefs

Cf. A088164, A228562.

Programs

Mathematica

Mod[Binomial[2#-1,#-1],#^3]&/@Select[Range[100],CompositeQ] (* Harvey P. Dale, May 03 2023 *)

PARI

forcomposite(c=1, 100, print1(lift(Mod(binomial(2*c-1, c-1), c^3)), ", "))

Extensions

Edited by Felix Fröhlich, May 27 2021