A356707 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y positive.

2, 3, 0, 2, 0, 0, 1, 4, 2, 2, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 1, 1, 1, 0, 2, 1, 0, 2, 0, 0, 0, 4, 2, 1, 1, 2, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 2, 3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 1, 2, 8, 0, 0, 0, 0, 2, 1, 4, 0, 1, 0, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 0, 1, 0, 0, 3, 1, 2

Offset

1,1

Comments

Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3 apart from the trivial solution (-n,0).

Links

Table of n, a(n) for n=1..100.

Formula

a(n) = (A081119(n^3)-1)/2 = (A356706(n)-1)/2 = A356706(n) - A356708(n).

Example

a(2) = 3 because the solutions to y^2 = x^3 + 2^3 with y > 0 are (1,3), (2,4), and (46,312).

Prog

(SageMath) [(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))-1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022

Crossrefs

Cf. A081119, A356706, A356708.

Indices of 0, 1, 2, and 3: A356709, A356710, A356711, A356712.

Sequence in context: A322147 A059066 A059067 * A352938 A065861 A329393

Adjacent sequences: A356704 A356705 A356706 * A356708 A356709 A356710

Keyword

nonn,hard

Author

Jianing Song, Aug 23 2022

Extensions

Offset and a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 03 2022

a(61)-a(100) from Max Alekseyev, Jun 01 2023

Status

approved