A266003 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.

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, 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, 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

Offset: 0

Author

Zhi-Wei Sun, Dec 19 2015

Keywords

• nonn

Comments

Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are nonnegative integers.

I have verified this for all integers m with |m| <= 10^5.

See also A266004 for a related sequence.

Examples

			a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
a(67) = 6337 since 67 = 676^4 - 6337^3 + 213662^2.
a(176) = 13449 since 176 = 140^4 - 13449^3 + 1559555^2.
a(2667) = 661^4 - 15655^3 + 1909401^2.
a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Checking the conjecture for m = 0..10^5

Crossrefs

Cf. A000290, A000578, A000583, A262827, A266004.

Programs

Mathematica

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