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

0,4

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.

Links

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

Example

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.

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]

Crossrefs

Cf.A000578, A001014, A351306, A351338.

Sequence in context: A040427 A070988 A358268 * A040426 A352137 A371094

Adjacent sequences: A351318 A351319 A351320 * A351322 A351323 A351324

Keyword

nonn

Author

Zhi-Wei Sun, Feb 07 2022

Status

approved