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%I #16 Aug 08 2020 01:33:20
%S 1,-3,4,9,-10,-12,16,21,25,37,-47,-48,49,64,-75,-87,88,100,105,121,
%T 134,-147,169,172,-192,196,-241,-243,256,289,-300,361,-363,400,443,
%U 484,-507,529,541,-588,625,676,-699,732,-759,-768,777,784,841,-867,897,961
%N Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 1458.
%C Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).
%C (11 + 3*n - 9*n^2)^3 + (11 + 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 1)^6 = 1458, the numbers of the form (3*n + 1)^2 are terms of the sequence.
%C (11 - 3*n - 9*n^2)^3 + (11 - 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 2)^6 = 1458, the numbers of the form (3*n + 2)^2 are also terms of the sequence.
%C Thus, A001651(n)^2 are terms of the sequence. There is an infinity of nontrivial solutions to the equation.
%D R. K. Guy, Unsolved Problems in Number Theory, D5.
%e 5^3 + 11^3 + 2 * 1^3 = 1458, 1 is a term.
%e (-1)^3 + (11)^3 + 2 * (4)^3 = 1458, 4 is a term.
%t Clear[t]
%t t = {};
%t Do[y = (1458 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -980, 980}, {x, -25319, 25319}]
%t u = Union@t;
%t v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 2000}];
%t Select[v, MemberQ[u, #] &]
%Y Cf. A000290, A000578, A001651, A003215, A004825, A004826, A050791, A130472, A195006.
%K sign
%O 1,2
%A _XU Pingya_, Jul 17 2020
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