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 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). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified January 27 21:49 EST 2023. Contains 359849 sequences. (Running on oeis4.)