A268303 Composite numbers n such that Sum_{k = 0..n} (-1)^k * C(n,k) * C(2*n,k) == -1 (mod n^3) (see A234839).

10, 25, 146, 586, 2186, 2386, 2594, 2642, 4162, 4226, 4258, 5186, 7745, 8258, 8354, 8458, 8714, 8746, 8842, 10306, 10378, 10786, 10826, 10834, 10898, 16418, 16546, 16706, 17026, 17674, 20546, 20642, 20738, 32834, 32906, 33322, 33505, 33802, 34058, 35338

Offset: 1

Author

Michel Marcus, Jan 31 2016

Keywords

• nonn

Comments

A234839(p) == -1 (mod p^3) for all primes >= 5. But some composites also satisfy this property. They are the object of this sequence.

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.

Links

• Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672. See Table 1 p. 2666.

Crossrefs

Cf. A234839.

Programs

PARI

isok(n) = Mod(sum(k=0, n, (-1)^k*binomial(n,k)*binomial(2*n,k)), n^3) == Mod(-1, n^3);
lista(nn) = forcomposite(n=2, nn, if (isok(n), print1(n, ", ")));