

A323488


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


1



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
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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^28)*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(k2)==2  k==5  ellanalyticrank(ellinit([0, k*k8, 0, 16, 0]))[1]; \\ Jinyuan Wang, Oct 11 2020


CROSSREFS

Sequence in context: A115945 A084750 A092027 * A199134 A131855 A102858
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



