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A254639 Least positive integer m such that A254631(m) = n.

%I A254639 #6 Feb 04 2015 04:36:17
%S A254639 2,1,6,16,27,62,71,92,122,161,176,216,286,386,351,491,577,492,781,866,
%T A254639 1023,617,736,1002,1504,1441,1402,1297,1451,1562,1842,2166,1682,1331,
%U A254639 2626,2311,2332,2969,3177,2761,2876,3641,3261,3697,3586,4894,3576,3921,4482,4542
%N A254639 Least positive integer m such that A254631(m) = n.
%C A254639 Conjecture: a(n) exists for any n > 0. Moreover, no term a(n) is divisible by 5.
%C A254639 It seems that no term a(n) is congruent to 8 modulo 10.
%H A254639 Zhi-Wei Sun, <a href="/A254639/b254639.txt">Table of n, a(n) for n = 1..160</a>
%H A254639 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 2009-2015.
%e A254639 a(3) = 6 since 6 is the least positive integer m with A254631(m) = 3. Note that 6 = 0*1/2 + 1*(3*1+2) + 1*(3*1-2) = 1*2/2 + 1*(3*1+2) + 0*(3*0-2) = 3*4/2 + 0*(3*0+2) + 0*(3*0-2).
%t A254639 TQ[n_]:=IntegerQ[Sqrt[8n+1]]
%t A254639 Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-y(3y+2)-z(3z-2)],r=r+1;If[r>n, Goto[aa]]],{y,0,(Sqrt[3m+1]-1)/3},{z,0,(Sqrt[3(m-y(3y+2))+1]+1)/3}];
%t A254639 If[r==n,Print[n, " ", m];Goto[bb],Goto[aa]]];Label[bb];Continue,{n,1,50}]
%Y A254639 Cf. A000217, A000567, A045944, A254573, A254574, A254595, A254617, A254631.
%K A254639 nonn
%O A254639 1,1
%A A254639 _Zhi-Wei Sun_, Feb 04 2015