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%I #23 Aug 08 2020 01:33:08
%S 1,4,9,25,49,81,121,169,-224,225,289,361,-383,441,504,529,625,729,841,
%T 961,1089,1225,1369,1521,1681,1849,2025,2209,-2219,2401,2601,-2687,
%U 2809,3025,3249,3481,-3680,3721,3969,4225,-4283,4417,4489,4761,5041,5329,-5459
%N Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 128.
%C Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).
%C (5 - 4*n^2)^3 + (5 - 4*(n + 1)^2)^3 + 2*(2*n + 1)^6 = 128. A000290(2*n + 1) are terms of the sequence, i.e., there is an infinity of nontrivial solutions to the equation.
%D R. K. Guy, Unsolved Problems in Number Theory, D5.
%e 1^3 + 5^3 + 2 * 1^3 = 128, 1 is a term.
%e (-11)^3 + (-31)^3 + 2 * (25)^3 = 128, 25 is a term.
%t Clear[t]
%t t = {};
%t Do[y = (128 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -4761, 4761}, {x, -11550, 11550}]
%t u = Union@t;
%t v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 9523}];
%t Select[v, MemberQ[u, #] &]
%Y Cf. A000290, A000578, A003215, A004825, A004826, A050791, A130472, A195006.
%K sign
%O 1,2
%A _XU Pingya_, Jul 12 2020
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