cp's OEIS Frontend

A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

%I A338933 #31 Apr 16 2022 05:40:12
%S A338933 2,16,128,1024,1458,8192,11664,31250,65536,93312,235298,524288,746496,
%T A338933 1062882,2000000,3543122,3906250,5971968,9653618,15059072,22781250,
%U A338933 28697814,33554432,47775744,48275138,68024448,80707214,94091762,128000000,171532242,226759808
%N A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.
%C A338933 The data are derived from the following formula:
%C A338933 (a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
%C A338933 (a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
%C A338933 (9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
%C A338933 (6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
%C A338933 (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 A338933 (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 A338933 (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 A338933 R. K. Guy, Unsolved Problems in Number Theory, D5.
%H A338933 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 A338933 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 A338933 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 A338933 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 A338933 t1 = 2*Range[23]^6;
%t A338933 t2 = 2*{1, 2, 4, 5, 7, 8}^9;
%t A338933 t3 = 1458*Range[4]^9;
%t A338933 t4 = 2*{1, 5}^12;
%t A338933 t5 = 16*{1, 2, 4}^12;
%t A338933 t6 = 1458*{1, 3}^12;
%t A338933 t7 = 11664*{1, 2, 3}^12;
%t A338933 Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]
%Y A338933 Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.
%K A338933 nonn
%O A338933 1,1
%A A338933 _XU Pingya_, Nov 16 2020
%E A338933 Missing terms 1024 and 746496 added by _XU Pingya_, Mar 14 2022