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.
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
KEYWORD
sign
AUTHOR
XU Pingya, Aug 08 2020
STATUS
approved
