A351321 Least positive integer m such that m^6*n = u^6 + v^3 - (x^6 + y^3) for some nonnegative integers u,v,x,y with x^6 + y^3 <= m^6*n^2.

1, 1, 1, 21, 6, 3, 1, 1, 1, 1, 7, 7, 3, 3, 3, 3, 2, 6, 1, 1, 1, 2, 6, 3, 5, 1, 1, 1, 1, 5, 2, 6, 12, 3, 1, 1, 1, 1, 1, 6, 6, 3, 3, 2, 1, 1, 4, 3, 2, 3, 3, 2, 7, 1, 1, 1, 1, 1, 3, 6, 1, 1, 1, 1, 1, 1, 4, 15, 3, 3, 1, 1, 1, 3, 4, 2, 3, 6, 3, 3, 2, 3, 1, 1, 3, 3, 3, 6, 1, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1

Offset: 0

Author

Zhi-Wei Sun, Feb 07 2022

Keywords

• nonn

Comments

6-6-3-3 Conjecture: Each rational number can be written as u^6 - v^6 + x^3 - y^3 with u,v,x,y nonnegative rational numbers. Moreover, a(n) exists for any nonnegative integer n.

As a/b = (a*b^5)/b^6 for any integer a and nonzero integer b, the second assertion in the conjecture implies the first one.

Examples

			a(3) = 21 with 21^6*3 = 22^6 + 956^3 - (30^6 + 93^3) and 30^6 + 93^3 <= 21^6*3^2.
a(67) = 15 with 15^6*67 = 21^6 + 1091^3 - (15^6 + 848^3) and 15^6 + 848^3 <= 15^6*67^2.
a(564) = 14 with 14^6*564 = 69^6 + 4415^3 - (16^6 + 5746^3) and 16^6 + 5746^3 <= 14^6*564^2.
a(949) = 18 with 18^6*949 = 7^6 + 11784^3 - (11^6 + 11706^3) and 11^6 + 11706^3 <= 18^6*949^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..1000

Crossrefs

Cf.A000578, A001014, A351306, A351338.

Programs

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[m=1;Label[bb];k=m^6;Do[If[CQ[k*n+x^6+y^3-z^6],tab=Append[tab,m];Goto[aa]],
{x,0,m*n^(1/3)},{y,0,(k*n^2-x^6)^(1/3)},{z,0,(k*n+x^6+y^3)^(1/6)}]; m=m+1;Goto[bb]; Label[aa],{n,0,100}];Print[tab]