A260935 Smallest k such that A261029(k) = n.

0, 1, 8, 28, 108, 189, 324, 648, 972, 756, 1701, 2457, 1512, 3888, 2268, 4536, 6048, 13104, 10584, 15120, 6804, 16848, 9072, 14364, 9828, 28728, 19656, 21168, 36288, 31752, 50544, 27216, 46683, 70308, 29484, 57456, 39312, 81648, 111132, 63504, 58968, 108864

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Aug 22 2015

Keywords

• nonn

Comments

Theorem. For every n>=0, a(n) exists.

Are all terms from a(4)=108 onward divisible by 9?

a(139) = 12006176 is not divisible by 9. - Chai Wah Wu, Aug 25 2015

Examples

			By condition z>=x+1>=1. By induction one can prove that F(x,y,z)>=3*z-2 (cf.[Shevelev]).
Since F>=1, then A261029(0)=0 and a(0)=0;
Further,
x y z F
0 0 1 1
0 1 1 2
Since F(x,y,2)>=4>1, A261029(1)=1 and a(1)=1.
0 0 2 8
0 1 2 9
0 2 2 16
1 1 2 4
1 2 2 5
0 0 3 27
0 1 3 28
0 2 3 35
0 3 3 54
1 1 3 20
1 2 3 18
1 3 3 28
2 2 3 7
2 3 3 8
Since F(x,y,4)>=10>8, A261029(8)=2 and a(2)=8,
etc.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..698
• Vladimir Shevelev, Representation of positive integers by the form x^3+y^3+z^3-3xyz, arXiv:1508.05748 [math.NT], 2015.
• Robert G. Wilson v, For n: solutions of A261029(k).

Crossrefs

Cf. A261029.

Programs

Mathematica

r[n_] := Reduce[0 <= x <= y <= z && z >= x + 1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
a29[n_] := a29[n] = Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
a[n_] := For[k=0, True, k++, If[a29[k] == n, Print[n, " ", k]; Return[k]]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 15 2018 *)

Formula

A261029(a(n)) = n.

For n>=1, a(n) <= 8^(n-1).

Extensions

a(11)-a(41) from Chai Wah Wu, Aug 25 2015