%I #9 Feb 22 2022 08:29:28

%S 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 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 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 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 Conjecture: a(n) exists for each nonnegative integer n.

%C See also Conjecture 2 in A351341.

%H Zhi-Wei Sun, <a href="/A351832/b351832.txt">Table of n, a(n) for n = 0..10000</a>

%e a(170) = 2730 with 170 = 9^6 + 15^6 - (2114^2 + 2730^2).

%e a(5938) = 16184 with 5938 = 17^6 + 25^6 - (2520^2 + 16184^2).

%e a(9746) = 7600 with 9746 = 11^6 + 21^6 - (5456^2 + 7600^2).

%t QQ[n_]:=IntegerQ[n^(1/6)];

%t 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 Cf. A000290, A001014, A001481, A351321, A351338, A351341, A351375, A351376.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Feb 21 2022