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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Dec 19 2015

Keywords

• nonn

Comments

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

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

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A000578, A000583, A266003.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
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}]