A266364 Least positive integer x such that n + x^4 = y^2 + z^3 for some positive integers y and z, or 0 if no such x exists.

6, 1, 69, 7, 1, 46, 13, 5, 1, 1, 2, 1, 2, 4, 27, 2, 1, 2, 28, 3, 2, 2, 37, 1, 4, 1, 11, 1, 2, 5, 1, 5, 1, 4, 2, 1, 1, 8, 4, 6, 8, 2, 1, 1, 6, 3, 3, 2, 3, 1, 18, 1, 2, 3, 6, 9, 1, 2, 6, 5, 2

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 A266363 for similar sequences.

Examples

			a(0) = 6 since 0 + 6^4 = 28^2 + 8^3.
a(2) = 69 since 2 + 69^4 = 44^2 + 283^3.
a(5) = 46 since 5 + 46^4 = 1742^2 + 113^3.
a(570) = 983 since 570 + 983^4 = 546596^2 + 8595^3.
a(8078) = 2255 since 8078 + 2255^4 = 1926054^2 + 28083^3.
		

Links

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

Crossrefs

Cf. A000290, A000578, A000583, A266152, A266153, A266230, A266231, A266277, A266314, A266363.

Programs

Mathematica

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