A351338 Least nonnegative integer m such that n = x^3 + y^3 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

0, 0, 0, 5, 11, 4, 1, 1, 0, 0, 3, 2, 2, 35, 1, 1, 0, 7, 2, 2, 2, 12, 14, 10, 4, 1, 1, 0, 0, 3, 3, 44, 22, 1, 1, 0, 3, 3, 2, 8, 8, 127, 4, 7, 3, 2, 2, 8, 2, 2, 97, 7, 1, 1, 0, 2, 2, 2, 17, 13, 4, 4, 1, 1, 0, 0, 6, 20, 4, 4, 1, 1, 0, 15, 3, 2, 53, 22, 7, 3, 4, 6, 2, 2, 5, 14, 139, 4, 4, 1, 1, 0, 5, 3, 5, 22, 4, 3, 3, 3, 3

Offset: 0

Author

Zhi-Wei Sun, Feb 08 2022

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n >= 0. Equivalently, each integer can be written as x^3 + y^3 - (z^3 + w^3) with x,y,z,w nonnegative integers.

This is stronger than Sierpinski's conjecture which states that any integer is a sum of four integer cubes.

Examples

			a(41) = 127 with 41 = 41^3 + 128^3 - 49^3 -127^3.
a(130) = 143 with 130 = 37^3 + 169^3 - 125^3 - 143^3.
a(4756) = 533 with 4756 = 265^3 + 538^3 - 284^3 - 533^3.
a(5134) = 389 with 5134 = 19^3 + 418^3 - 242^3 - 389^3.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000578, A004826, A004999, A351306, A351321.

Programs

Mathematica

CQ[n_]:=IntegerQ[n^(1/3)];
tab={};Do[m=0; Label[bb]; k=m^3; Do[If[CQ[n+k+x^3-y^3], tab=Append[tab,m];Goto[aa]],  {x, 0, m}, {y, 0, ((n+k+x^3)/2)^(1/3)}];m=m+1; Goto[bb]; Label[aa], {n, 0, 100}];Print[tab]