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%I #19 Sep 22 2025 08:25:30
%S 1,2,3,4,5,8,9,10,12,13,14,15,16,17,18,19,20,21,22,25,27,29,30,32,33,
%T 34,35,36,37,39,40,41,43,45,46,48,49,50,51,52,53,56,57,58,59,60,62,64,
%U 65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,86,87,88
%N Numbers k such that Mordell's equation y^2 = x^3 - k^3 has exactly 1 integral solution.
%C Numbers k such that Mordell's equation y^2 = x^3 - k^3 has no solution other than the trivial solution (k,0).
%C Cube root of A179163.
%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), so y^2 = x^3 - t^6 has only one solution (t^2,0).
%H Jianing Song, <a href="/A356713/b356713.txt">Table of n, a(n) for n = 1..108</a> (using data from A179149)
%e 1 is a term since the equation y^2 = x^3 - 1^3 has no solution other than (1,0).
%Y Cf. A081120, A179163, A356709, A356720. Complement of A228948.
%K nonn
%O 1,2
%A _Jianing Song_, Aug 23 2022