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%I #27 Jan 05 2021 05:05:42
%S 1,2,128,729,1458,4096,65536,93312,2985984,3906250,16777216,28697814,
%T 33554432,47775744,80707214,244140625,250000000,387420489,1836660096,
%U 2847656250,4715895382,5165261696,12230590464,13841287201,17179869184,21208998746,24461180928
%N Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.
%C The data are derived from the following formula:
%C (a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;
%C (4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;
%C (9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;
%C (36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;
%C ((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;
%C ((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.
%D R. K. Guy, Unsolved Problems in Number Theory, D5.
%H Kenji Koyama, <a href="https://doi.org/10.1090/S0025-5718-00-01202-3">On searching for solutions of the Diophantine equation x^3 + y^3 + 2z^3 = n</a>, Math. Comp, 69 (2000), 1735-1742.
%H J. C. P. Miller & M. F. C. Woollett, <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=67916">Solutions of the Diophantine equation x^3 + y^3 + z^3 = k</a>, J. London Math. Soc. 30(1955), 101-110.
%H Beniamino Segre, <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=46064">On the rational solutions of homogeneous cubic equations in four variables</a>, Math. Notae, 11 (1951), 1-68.
%e 128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.
%t t1 = 2*{1, 5, 7, 11, 13}^9;
%t t2 = 128*{1, 2, 4, 5, 7, 8}^9;
%t t3 = 1458*{1, 3, 5, 7, 9}^9;
%t t4 = 93312*{1, 2, 3, 4, 5}^9;
%t t5 = {1, 2, 4, 5, 7}^12;
%t t6 = 729*{1, 2, 3, 4, 5}^12;
%t Take[Union[t1, t2, t3, t4, t5, t6], 27]
%Y Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338933.
%K nonn
%O 1,2
%A _XU Pingya_, Nov 16 2020
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