%I #8 Feb 20 2022 23:06:38

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

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

%H Zhi-Wei Sun, <a href="/A351338/b351338.txt">Table of n, a(n) for n = 0..10000</a>

%e a(41) = 127 with 41 = 41^3 + 128^3 - 49^3 -127^3.

%e a(130) = 143 with 130 = 37^3 + 169^3 - 125^3 - 143^3.

%e a(4756) = 533 with 4756 = 265^3 + 538^3 - 284^3 - 533^3.

%e a(5134) = 389 with 5134 = 19^3 + 418^3 - 242^3 - 389^3.

%t CQ[n_]:=IntegerQ[n^(1/3)];

%t 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]

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

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Feb 08 2022