A336449 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*4^6.

1, 9, 16, 25, 49, 81, 121, 169, 225, 289, -356, 361, 441, 529, 625, 729, 841, -948, 961, 1045, 1089, 1225, 1369, 1521, 1681, -1715, 1849, 1876, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, -3587, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5769, 5929

Offset

1,2

Comments

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

Let x = a^(2*k) - (a^k)*t - t^2, y = a^(2*k) + (a^k)*t - t^2, z = t^2; then x^3 + y^3 + 2*z^3 = 2*a^(6*k). When a = 4, k = 1, t = 2*n + 1; (x, y, z) are primitive solutions of equation. Thus, terms of A016754 are terms of the sequence.

References

R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Table of n, a(n) for n=1..47.

Example

(-15)^3 + (-27)^3 + 2*25^3 = 11^3 + (-29)^3 + 2*25^3 = 8192, 25 is a term.
(-65)^3 + (449)^3 + 2*(-356)^3 = 8192, -356 is a term.

Mathematica

Clear[t]
t = {};
Do[y = (8192 - x^3 - 2z^3)^(1/3) /. (-1)^(1/3) -> -1;
If[Abs@x <= Abs@y && IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -5929, 5929}, {x, -Round[(Abs[8192 - 2z^3]/3)^(1/2)], Round[(Abs[8192 - 2z^3]/3)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 12000}];
Select[v, MemberQ[u, #] &]

Crossrefs

Cf. A000290, A000578, A003215, A004825, A004826, A016754, A050791, A130472, A195006, A336450.

Sequence in context: A256705 A092941 A075393 * A161183 A183345 A303873

Adjacent sequences: A336446 A336447 A336448 * A336450 A336451 A336452

Keyword

sign

Author

XU Pingya, Aug 08 2020

Status

approved