A266230 Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

3, 1, 3703, 5, 43, 2, 119, 3, 1, 19, 5, 384, 2, 29, 29, 1, 7, 18, 6, 3, 13, 14, 869, 7, 2, 15, 3, 1, 10, 5, 23, 2, 20, 10, 1, 45, 6, 2373, 4, 1193, 5, 52, 7, 36, 54, 3, 18, 5, 13, 4, 2, 385, 9, 1, 14, 6, 3, 76, 250, 250, 34, 2, 8, 3, 1, 336, 5, 52, 2, 8, 28, 1, 21, 12, 13, 4, 113

Offset: 0

Author

Zhi-Wei Sun, Dec 24 2015

Keywords

• nonn

Comments

Conjecture: For any integer m, there are positive integers x, y and z such that m + x^2 = y^3 + z^3.

This is similar to the conjecture in A266152. We have verified it for all integers m with |m| <= 25000.

Obviously, a(k^3) = 1 for any positive integer k.

See also A266231 for a related sequence.

Examples

			a(0) = 3 since 0 + 3^2 = 1^3 + 2^3.
a(2) = 3703 since 2 + 3703^2 = 107^3 + 232^3.
a(3) = 5 since 3 + 5^2 = 1^3 + 3^3.
a(4) = 43 since 4 + 43^2 = 5^3 + 12^3.
a(37) = 2373 since 37 + 2373^2 = 93^3 + 169^3.
a(1227) = 132316 since 1227 + 132316^2 = 1874^3 + 2219^3.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Checking the conjecture for integers m with 10000 < |m| <= 25000

Crossrefs

Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266231.

Programs

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,80}]