A323488 Nonnegative integers of the form x + 1/x + y + 1/y for some rational numbers x, y.

0, 4, 5, 10, 11, 13, 15, 17, 18, 19, 21, 25, 28, 29, 31, 33, 37, 38, 40, 43, 44, 47, 48, 50, 54, 56, 57, 58, 59, 61, 63, 65, 66, 68, 70, 71, 74, 75, 76, 79, 83, 86, 87, 88, 89, 91, 92, 93, 97, 102, 105, 106, 107, 108, 110, 112, 114, 115, 116, 119, 120, 122

Offset

1,2

Comments

Nonnegative integers k such that the elliptic curve given by the equation y^2 + x*y + y/k^2 = x^3 + x^2/k^2 has more than 4 rational points. (Note that if k is 0 or 4 the equation does not define an elliptic curve.)

If (X, Y) is a rational point on Y^2 = X^3 + (k^2-8)*X^2 + 16*X with k > 4 and (X, Y) not being a torsion point, then k = x + 1/x + y + 1/y, where x = 2*(Y - k*X)/(X^2 - 4*X) and y = (Y - k*X)/(8 - 2*X). - Jinyuan Wang, Oct 11 2020

Links

Jinyuan Wang, Table of n, a(n) for n = 1..2000

Example

17 is in the sequence since x=80/3 and y=-5/48 give the solution: 80/3 + 3/80 - 5/48 - 48/5 = 17.

Prog

(PARI) is(k) = abs(k-2)==2 || k==5 || ellanalyticrank(ellinit([0, k*k-8, 0, 16, 0]))[1]; \\ Jinyuan Wang, Oct 11 2020

Crossrefs

Sequence in context: A115945 A092027 A084750 * A199134 A369654 A131855

Adjacent sequences: A323485 A323486 A323487 * A323489 A323490 A323491

Keyword

nonn

Author

Xavier Xarles, Aug 30 2019

Extensions

Name edited by Jinyuan Wang, Oct 11 2020

Status

approved