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

1,1

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

Table of n, a(n) for n=1..40.

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.

Prog

(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, ", ")));

Crossrefs

Cf. A234839.

Sequence in context: A251310 A251194 A071289 * A248833 A220039 A219377

Adjacent sequences: A268300 A268301 A268302 * A268304 A268305 A268306

Keyword

nonn

Author

Michel Marcus, Jan 31 2016

Status

approved