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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.

%I A351338 #8 Feb 20 2022 23:06:38
%S A351338 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 A351338 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 A351338 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 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.
%C A351338 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 A351338 This is stronger than Sierpinski's conjecture which states that any integer is a sum of four integer cubes.
%H A351338 Zhi-Wei Sun, <a href="/A351338/b351338.txt">Table of n, a(n) for n = 0..10000</a>
%e A351338 a(41) = 127 with 41 = 41^3 + 128^3 - 49^3 -127^3.
%e A351338 a(130) = 143 with 130 = 37^3 + 169^3 - 125^3 - 143^3.
%e A351338 a(4756) = 533 with 4756 = 265^3 + 538^3 - 284^3 - 533^3.
%e A351338 a(5134) = 389 with 5134 = 19^3 + 418^3 - 242^3 - 389^3.
%t A351338 CQ[n_]:=IntegerQ[n^(1/3)];
%t A351338 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 A351338 Cf. A000578, A004826, A004999, A351306, A351321.
%K A351338 nonn
%O A351338 0,4
%A A351338 _Zhi-Wei Sun_, Feb 08 2022