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

%I A260935 #66 Dec 15 2018 20:43:41
%S A260935 0,1,8,28,108,189,324,648,972,756,1701,2457,1512,3888,2268,4536,6048,
%T A260935 13104,10584,15120,6804,16848,9072,14364,9828,28728,19656,21168,36288,
%U A260935 31752,50544,27216,46683,70308,29484,57456,39312,81648,111132,63504,58968,108864
%N A260935 Smallest k such that A261029(k) = n.
%C A260935 Theorem. For every n>=0, a(n) exists.
%C A260935 Are all terms from a(4)=108 onward divisible by 9?
%C A260935 a(139) = 12006176 is not divisible by 9. - _Chai Wah Wu_, Aug 25 2015
%H A260935 Chai Wah Wu, <a href="/A260935/b260935.txt">Table of n, a(n) for n = 0..698</a>
%H A260935 Vladimir Shevelev, <a href="http://arxiv.org/abs/1508.05748">Representation of positive integers by the form x^3+y^3+z^3-3xyz</a>, arXiv:1508.05748 [math.NT], 2015.
%H A260935 Robert G. Wilson v, <a href="/A260935/a260935.txt">For n: solutions of A261029(k).</a>
%F A260935 A261029(a(n)) = n.
%F A260935 For n>=1, a(n) <= 8^(n-1).
%e A260935 By condition z>=x+1>=1. By induction one can prove that F(x,y,z)>=3*z-2 (cf.[Shevelev]).
%e A260935 Since F>=1, then A261029(0)=0 and a(0)=0;
%e A260935 Further,
%e A260935 x y z F
%e A260935 0 0 1 1
%e A260935 0 1 1 2
%e A260935 Since F(x,y,2)>=4>1, A261029(1)=1 and a(1)=1.
%e A260935 0 0 2 8
%e A260935 0 1 2 9
%e A260935 0 2 2 16
%e A260935 1 1 2 4
%e A260935 1 2 2 5
%e A260935 0 0 3 27
%e A260935 0 1 3 28
%e A260935 0 2 3 35
%e A260935 0 3 3 54
%e A260935 1 1 3 20
%e A260935 1 2 3 18
%e A260935 1 3 3 28
%e A260935 2 2 3 7
%e A260935 2 3 3 8
%e A260935 Since F(x,y,4)>=10>8, A261029(8)=2 and a(2)=8,
%e A260935 etc.
%t A260935 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];
%t A260935 a29[n_] := a29[n] = Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
%t A260935 a[n_] := For[k=0, True, k++, If[a29[k] == n, Print[n, " ", k]; Return[k]]];
%t A260935 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 15 2018 *)
%Y A260935 Cf. A261029.
%K A260935 nonn
%O A260935 0,3
%A A260935 _Vladimir Shevelev_ and _Peter J. C. Moses_, Aug 22 2015
%E A260935 a(11)-a(41) from _Chai Wah Wu_, Aug 25 2015