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A266004 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. 6

%I #15 Jun 27 2017 14:56:39

%S 1,3,2,2,13,2,2,2,5,3,3,4,15,4,4,33,3,3,5,6,31,3,3,5,3,3,3,4,21,11,6,

%T 4,17,11,5,98,7,4,4,5,147,19,5,4,5,6,4,4,65,1011,7,9,7,4,4,6,59,47,4,

%U 4,5,4,4,4,13,7,18,9,175,8,6,6,5,15,5,5,103,7,6,13,11,27,7,5,375,6,7,5,5,11,13,13,5,6,6,8,413,379,5,5

%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 The conjecture in A266003 implies that a(n) > 0 for all n > 0.

%C It seems that a(n) <= n^2 for any n > 0.

%H Zhi-Wei Sun, <a href="/A266004/b266004.txt">Table of n, a(n) for n = 1..10000</a>

%e a(50) = 1011 since -50 = 78^4 - 1011^3 + 31565^2.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t Do[y=Ceiling[n^(1/3)];Label[bb];Do[If[SQ[y^3-n-x^4],Goto[aa]],{x,0,(n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Print[n," ",y];Continue,{n,1,100}]

%Y Cf. A000290, A000578, A000583, A266003.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Dec 19 2015

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