%I #16 Jun 06 2023 17:40:44

%S 1,4,9,10,14,16,25,28,33,36,37,40,49,64,70,81,84,88,90,91,100,104,121,

%T 126,130,132,140,144,154,160,169,176,184,193,196

%N Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

%C Cube root of A179149.

%C Contains all squares: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3).

%e 1 is a term since the equation y^2 = x^3 + 1^3 has 5 solutions (-1,0), (0,+-1), and (2,+-3).

%Y Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356712.

%Y Indices of 5 in A356706, of 2 in A356707, and of 3 in A356708.

%K nonn,hard,more

%O 1,2

%A _Jianing Song_, Aug 23 2022

%E a(31)-a(35) from _Max Alekseyev_, Jun 01 2023