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Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y nonnegative.
7

%I #21 Jun 02 2023 01:56:41

%S 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,

%T 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,

%U 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

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

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

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

%e 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).

%o (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

%Y Cf. A081119, A134108, A356706, A356707.

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

%K nonn,hard

%O 1,1

%A _Jianing Song_, Aug 23 2022

%E a(21) corrected and a(22)-a(60) by _Lucas A. Brown_, Sep 04 2022

%E a(61)-a(100) from _Max Alekseyev_, Jun 01 2023