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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 (list; graph; refs; listen; history; text; internal format)
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 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

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Last modified August 10 13:12 EDT 2022. Contains 356039 sequences. (Running on oeis4.)