A266363 Least positive integer x such that n + x^2 = y^3 + z^4 for some positive integers y and z, or 0 if no such x exists.

3, 1, 302, 5, 47, 2, 362, 6, 1, 372, 14, 61, 4, 2, 70, 3, 1, 24, 5, 3, 2, 14, 364, 1, 2, 8, 10, 1, 454, 6, 848, 7, 15, 7, 3, 18, 14, 13, 1362, 2, 5, 10, 1, 37, 6, 9, 6, 68, 13, 4, 24, 36, 37, 6, 26, 5, 3, 5, 15, 7, 9

Offset: 0

Author

Zhi-Wei Sun, Dec 28 2015

Keywords

• nonn

Comments

The general conjecture in A266277 implies that for any integer m there are positive integers x, y and z such that m + x^2 = y^3 + z^4.

See also A266152 and A266364 for similar sequences.

Examples

			a(0) = 3 since 0 + 3^2 = 2^3 + 1^4.
a(2) = 302 since 2 + 302^2 = 45^3 + 3^4.
a(3) = 5 since 3 + 5^2 = 3^3 + 1^4.
a(38) = 1362 since 38 + 1362^2 = 121^3 + 17^4.
a(394) = 110307 since 394 + 110307^2 = 2283^3 + 128^4.
a(5546) = 945840 since 5546 + 945840^2 = 9625^3 + 233^4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000290, A000578, A000583, A266152, A266153, A266230, A266231, A266277, A266314, A266364.

Programs

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^4],Print[n," ",x];Goto[aa]],{y,1,(n+x^2-1)^(1/4)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,60}]