A266152 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.

8, 1, 2, 17, 1, 3, 139, 19, 37, 1, 3, 9, 2, 7, 3, 1411, 1, 2, 2, 1, 5, 4, 387, 3, 1, 1, 4, 7, 9, 2, 35, 1, 33, 2, 6, 5, 1, 4, 3, 11, 1, 6, 2, 429, 2, 5, 11, 179, 73, 1, 15, 1, 4, 3, 11, 3, 5, 2, 3, 15, 5, 6, 7, 3, 1, 6, 4, 6337, 8, 16, 3

Offset: 0

Author

Zhi-Wei Sun, Dec 22 2015

Keywords

• nonn

Comments

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

This is slightly stronger than the conjecture in A266003.

See also A266153 for a related sequence, and A266212 for a stronger conjecture.

If n is a positive square, then a(n) = 1. - Altug Alkan, Dec 23 2015

Examples

			a(0) = 8 since 0 = 4^4 - 8^3 + 16^2.
a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
a(15) = 1411 since 15 = 119^4 - 1411^3 + 51075^2.
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 integers m with |m| <= 10^5
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Crossrefs

Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266153, A266212.

Programs

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=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,0,70}]