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

%I A244214 #40 Jun 27 2023 15:07:54
%S A244214 35,30,291,253,378,782,2404,1260,291,3378,410,7899,3996,6030,126,
%T A244214 10988,11188,5180,19712,8483,5334,34394,1841,21410,20580,39556,38810,
%U A244214 64260,35972,66060,36504,61326,1716,123628,49140,63748,124392,20091,99388,157767,24392
%N A244214 a(n) = binomial(2*c-1, c-1) (mod c^3), where c is the n-th composite.
%C A244214 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).
%H A244214 Felix Fröhlich, <a href="/A244214/b244214.txt">Table of n, a(n) for n = 1..10000</a>
%H A244214 C. Helou and G. Terjanian, <a href="https://doi.org/10.1016/j.jnt.2007.06.008">On Wolstenholme's theorem and its converse</a>, Journal of Number Theory, Volume 128, Issue 3 (2008), 475-499.
%H A244214 R. J. McIntosh, <a href="https://doi.org/10.4064/aa-71-4-381-389">On the converse of Wolstenholme's Theorem</a>, Acta Arithmetica, 71 (1995), 381-389.
%H A244214 V. Trevisan and K. Weber, <a href="http://mc.sbm.org.br/docs/mc/pdf/21/a16.pdf">Testing the converse of Wolstenholme's theorem</a>, Matematica Contemporanea, 21 (2001), 275-286.
%H A244214 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>
%t A244214 Mod[Binomial[2#-1,#-1],#^3]&/@Select[Range[100],CompositeQ] (* _Harvey P. Dale_, May 03 2023 *)
%o A244214 (PARI) forcomposite(c=1, 100, print1(lift(Mod(binomial(2*c-1, c-1), c^3)), ", "))
%Y A244214 Cf. A088164, A228562.
%K A244214 nonn
%O A244214 1,1
%A A244214 _Felix Fröhlich_, Jun 23 2014
%E A244214 Edited by _Felix Fröhlich_, May 27 2021