A338933 - OEIS

%I #31 Apr 16 2022 05:40:12

%S 2,16,128,1024,1458,8192,11664,31250,65536,93312,235298,524288,746496,

%T 1062882,2000000,3543122,3906250,5971968,9653618,15059072,22781250,

%U 28697814,33554432,47775744,48275138,68024448,80707214,94091762,128000000,171532242,226759808

%N Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

%C The data are derived from the following formula:

%C (a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6

%C (a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;

%C (9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;

%C (6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;

%C (6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;

%C (18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;

%C (18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*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 16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.

%t t1 = 2*Range[23]^6;

%t t2 = 2*{1, 2, 4, 5, 7, 8}^9;

%t t3 = 1458*Range[4]^9;

%t t4 = 2*{1, 5}^12;

%t t5 = 16*{1, 2, 4}^12;

%t t6 = 1458*{1, 3}^12;

%t t7 = 11664*{1, 2, 3}^12;

%t Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]

%Y Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.

%K nonn

%O 1,1

%A _XU Pingya_, Nov 16 2020

%E Missing terms 1024 and 746496 added by _XU Pingya_, Mar 14 2022