A356707 - OEIS

%I #25 Jun 06 2023 15:36:10

%S 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,

%T 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,

%U 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

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

%C 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).

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

%e a(2) = 3 because the solutions to y^2 = x^3 + 2^3 with y > 0 are (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 03 2022

%Y Cf. A081119, A356706, A356708.

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

%K nonn,hard

%O 1,1

%A _Jianing Song_, Aug 23 2022

%E Offset and a(21) corrected and a(22)-a(60) by _Lucas A. Brown_, Sep 03 2022

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