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A351832 Least nonnegative integer m such that n = x^6 + y^6 - (z^2 + m^2) for some nonnegative integers x,y,z with z <= m.

%I A351832 #9 Feb 22 2022 08:29:28
%S A351832 0,0,0,6,6,20,7,7,19,24,24,7,6,6,5,5,7,26,26,6,6,22,9,5,5,6,98,6,6,6,
%T A351832 5,5,4,4,32,5,5,26,5,4,4,20,322,7,4,4,3,3,4,4,22,3,3,22,3,3,2,2,418,2,
%U A351832 2,2,1,1,0,0,94,6,23,20,7,19,24,20,20,7,6,22,5,7,19,18,18,6,22,37,59,5,6,24,24,6,6,21
%N A351832 Least nonnegative integer m such that n = x^6 + y^6 - (z^2 + m^2) for some nonnegative integers x,y,z with z <= m.
%C A351832 Conjecture: a(n) exists for each nonnegative integer n.
%C A351832 See also Conjecture 2 in A351341.
%H A351832 Zhi-Wei Sun, <a href="/A351832/b351832.txt">Table of n, a(n) for n = 0..10000</a>
%e A351832 a(170) = 2730 with 170 = 9^6 + 15^6 - (2114^2 + 2730^2).
%e A351832 a(5938) = 16184 with 5938 = 17^6 + 25^6 - (2520^2 + 16184^2).
%e A351832 a(9746) = 7600 with 9746 = 11^6 + 21^6 - (5456^2 + 7600^2).
%t A351832 QQ[n_]:=IntegerQ[n^(1/6)];
%t A351832 tab={};Do[m=0; Label[bb]; k=m^2; Do[If[QQ[n+k+x^2-y^6], tab=Append[tab,m];Goto[aa]],  {x, 0, m}, {y, 0, ((n+k+x^2)/2)^(1/6)}];m=m+1; Goto[bb]; Label[aa], {n, 0, 100}];Print[tab]
%Y A351832 Cf. A000290, A001014, A001481, A351321, A351338, A351341, A351375, A351376.
%K A351832 nonn
%O A351832 0,4
%A A351832 _Zhi-Wei Sun_, Feb 21 2022