

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
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OFFSET

1,1


COMMENTS

After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...)


LINKS

Table of n, a(n) for n=1..56.
JosephAlfred Serret, Section 512, Cours d'algĂ¨bre supĂ©rieure, GauthierVillars.


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



