A266153 Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

3, 3, 2, 6, 13, 2, 3, 5, 5, 3, 28, 4, 15, 4, 10, 33, 3, 7, 5, 238, 31, 3, 4, 5, 3, 11, 4, 5, 21, 11, 6, 4, 17, 11, 5, 98, 7, 4, 4, 5, 147, 19, 5, 4, 5, 6, 4, 29, 75, 1011, 7, 9, 7, 4, 8, 6, 59, 47, 4, 5, 71, 4, 17, 45, 13, 7, 18, 9, 175, 8

Offset: 1

Author

Zhi-Wei Sun, Dec 22 2015

Keywords

• nonn

Comments

The conjecture in A266152 implies that a(n) > 0 for all n > 0.

It seems that a(n) < n*(n+4)/2 for all n > 1.

Examples

			a(1) = 3 since -1 = 1^4 - 3^3 + 5^2.
a(2) = 3 since -2 = 2^4 - 3^3 + 3^2.
a(11) = 28 since -11 = 5^4 - 28^3 + 146^2.
a(20) = 238 since -20 = 32^4 - 238^3 + 3526^2.
		

Links

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

Crossrefs

Cf. A000290, A000578, A000583, A262827, A266004, A266152.

Programs

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=Floor[n^(1/3)]+1;Label[bb];Do[If[SQ[-n+y^3-x^4],Print[n," ",y];Goto[aa]],{x,1,(-n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Continue,{n,1,70}]