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 A258692 Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square. 1
 -4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This sequence is finite as there are finitely many integer solutions to the elliptic curve y^2 = x(x + 2)(x + 4) + 1 = x^3 + 6x^2 + 8x + 1. The x values of the integer solutions are {-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272}. This equation has more integer and natural number solutions than the equation that defines sequence A121234. LINKS EXAMPLE 1 * 3 * 5 + 1 = 16 = 4^2, so 4 is in the sequence. 2 * 4 * 6 + 1 = 49 = 7^2, so 2 is in the sequence. 3 * 5 * 7 + 1 = 106 = 2 * 53, so 3 is not in the sequence. MATHEMATICA Select[Range[-10, 100], IntegerQ[Sqrt[#(# + 2)(# + 4) + 1]] &] (* Alonso del Arte, Jun 12 2015 *) PROG (MAGMA) P := PolynomialRing(Integers()); {x: x in Sort([ p : p in IntegralPoints(EllipticCurve(n^3 + 6*n^2 + 8*n + 1)) ])}; (SageMath) [i for i in EllipticCurve([0, 6, 0, 8, 1]).integral_points()] # Seiichi Manyama, Aug 26 2019 CROSSREFS Cf. A121234. Sequence in context: A143543 A176863 A067017 * A067018 A200233 A327351 Adjacent sequences:  A258689 A258690 A258691 * A258693 A258694 A258695 KEYWORD sign,fini,full AUTHOR Morris Neene, Jun 12 2015 STATUS approved

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Last modified June 24 14:24 EDT 2021. Contains 345417 sequences. (Running on oeis4.)