A356708 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y nonnegative.

3, 4, 1, 3, 1, 1, 2, 5, 3, 3, 2, 1, 1, 3, 1, 3, 1, 4, 1, 1, 2, 2, 2, 1, 3, 2, 1, 3, 1, 1, 1, 5, 3, 2, 2, 3, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 3, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 2, 3, 9, 1, 1, 1, 1, 3, 2, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 1, 1, 3, 1, 2, 1, 3, 1, 3, 3, 2, 1, 1, 2, 1, 1, 4, 2, 3

Offset: 1

Author

Jianing Song, Aug 23 2022

Keywords

• nonn
• hard

Comments

Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3.

Examples

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

Crossrefs

Cf. A081119, A134108, A356706, A356707.

Indices of 1, 2, 3, and 4: A356709, A356710, A356711, A356712.

Programs

SageMath

[(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))+1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 04 2022

Formula

a(n) = (A081119(n^3)+1)/2 = A134108(n^3) = (A356706(n)+1)/2 = A356707(n)+1.

Extensions

a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 04 2022

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