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%I #8 Feb 01 2016 05:24:04
%S 10,25,146,586,2186,2386,2594,2642,4162,4226,4258,5186,7745,8258,8354,
%T 8458,8714,8746,8842,10306,10378,10786,10826,10834,10898,16418,16546,
%U 16706,17026,17674,20546,20642,20738,32834,32906,33322,33505,33802,34058,35338
%N Composite numbers n such that Sum_{k = 0..n} (-1)^k * C(n,k) * C(2*n,k) == -1 (mod n^3) (see A234839).
%C A234839(p) == -1 (mod p^3) for all primes >= 5. But some composites also satisfy this property. They are the object of this sequence.
%C It appears that these composite are semiprimes with one factor always 2 or 5. See "3. Composite solutions of (1.5)" section in Chamberland et al. link.
%H Marc Chamberland and Karl Dilcher, <a href="http://dx.doi.org/10.1016/j.jnt.2009.05.010">A Binomial Sum Related to Wolstenholme's Theorem</a>, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672. See Table 1 p. 2666.
%o (PARI) isok(n) = Mod(sum(k=0, n, (-1)^k*binomial(n,k)*binomial(2*n,k)), n^3) == Mod(-1, n^3);
%o lista(nn) = forcomposite(n=2, nn, if (isok(n), print1(n, ", ")));
%Y Cf. A234839.
%K nonn
%O 1,1
%A _Michel Marcus_, Jan 31 2016
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