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 A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails. 1
 3, 7, 9, 21, 21, 39, 61, 63, 93, 129, 169, 171, 219, 273, 331, 333, 399, 471, 547, 549, 633, 723, 817, 819, 921, 1029, 1141, 1143, 1263, 1389, 1519, 1521, 1659, 1803, 1951, 1953, 2109, 2271, 2437, 2439, 2613, 2793, 2977, 2979, 3171, 3369, 3571, 3573, 3783, 3999, 4219, 4221, 4449, 4683, 4921, 4923 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...) LINKS Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars. EXAMPLE For the first entry of Q=3, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are Q=7 with the polynomial x^3 - 7x + 7, then Q=9 with the polynomial x^3 - 9x + 9, and Q=21 with the polynomials x^3 - 21x + 35 and x^3 - 21x + 37. Note that for the Q=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921. MATHEMATICA SetOfQRs = {}; M = 1000; Do[   If[Divisible[3 (a^2 - a + 1), c^2] &&     Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&     3 (a^2 - a + 1)/c^2 <=  M,    SetOfQRs =     Union[SetOfQRs, { { (3 (a^2 - a + 1))/        c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],   {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}]; Print[SetOfQRs // MatrixForm]; CROSSREFS Cf. A316184. Contained in the union of A034017 and three times A034017. Sequence in context: A304539 A306124 A096102 * A045797 A118555 A056652 Adjacent sequences:  A316154 A316155 A316156 * A316158 A316159 A316160 KEYWORD nonn AUTHOR Greg Dresden, Jun 25 2018 EXTENSIONS More terms from Robert G. Wilson v, Jul 02 2018 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)