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%I #21 Dec 21 2015 08:30:17
%S 0,0,0,1,0,0,139,19,1,0,0,9,2,7,3,1,0,0,2,1,0,4,3,3,1,0,0,7,2,2,19,1,
%T 0,2,6,1,0,0,3,11,1,0,2,429,2,5,11,179,1,0,0,1,0,3,3,3,2,2,3,15,5,6,7,
%U 1,0,0,4,6337,8,16,3,5,2,2,2,31,6,2,11,1,0,0,0,17,1,0,11,5,18,1,0,621,2,2,3,3,1,0,2,1,0
%N Least nonnegative integer y such that n = x^4 - y^3 + z^2 for some nonnegative integers x and z, or -1 if no such y exists.
%C Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are nonnegative integers.
%C I have verified this for all integers m with |m| <= 10^5.
%C See also A266004 for a related sequence.
%H Zhi-Wei Sun, <a href="/A266003/b266003.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="/A266003/a266003.txt">Checking the conjecture for m = 0..10^5</a>
%e a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
%e a(67) = 6337 since 67 = 676^4 - 6337^3 + 213662^2.
%e a(176) = 13449 since 176 = 140^4 - 13449^3 + 1559555^2.
%e a(2667) = 661^4 - 15655^3 + 1909401^2.
%e a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[y=0;Label[bb];Do[If[SQ[n+y^3-x^4],Goto[aa]],{x,0,(n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Print[n," ",y];Continue,{n,0,100}]
%Y Cf. A000290, A000578, A000583, A262827, A266004.
%K nonn
%O 0,7
%A _Zhi-Wei Sun_, Dec 19 2015
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