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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.
6
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
0,4
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.
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 08 2022
STATUS
approved