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 A121234 Nonnegative integers k such that k*(k+1)*(k+2)+1 is a square. 2
 0, 2, 4, 55 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The sequence is finite by Thue's theorem n(n + 1)(n + 2) + 1 = n^3 + 3*n^2 + 2*n + 1. The set of k values of integral solutions to the elliptic curve y^2 = n^3 + 3*n^2 + 2*n + 1 (see Magma program) is { -2, -1, 0, 2, 4, 55 }. So the sequence is complete. - Mohamed Bouhamida, Nov 29 2007 LINKS EXAMPLE 2 * 3 * 4 + 1 = 25 = 5^2, so 2 is in the sequence. 4 * 5 * 6 + 1 = 121 = 11^2, so 4 is in the sequence. 6 * 7 * 8 + 1 = 337, which is a prime number, so 6 is not in the sequence. PROG (MAGMA) P := PolynomialRing(Integers()); {k: k in Sort([ p : p in IntegralPoints(EllipticCurve(n^3 + 3*n^2 + 2*n + 1)) ])}; // Mohamed Bouhamida, Nov 29 2007 (PARI) isok(k) = issquare(k*(k+1)*(k+2)+1);  \\ Altug Alkan, Dec 07 2015 CROSSREFS Cf. A258692. Sequence in context: A092389 A341733 A005274 * A084574 A087621 A062784 Adjacent sequences:  A121231 A121232 A121233 * A121235 A121236 A121237 KEYWORD fini,nonn,full AUTHOR Roger Cuculière, Aug 21 2006 STATUS approved

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Last modified May 13 23:41 EDT 2021. Contains 343868 sequences. (Running on oeis4.)