Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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