%I #10 Feb 03 2015 08:47:01
%S 49,1,3,20,8,15,30,22,58,48,93,78,92,148,113,127,155,198,197,323,268,
%T 272,288,345,358,338,555,568,443,498,612,547,653,730,708,687,722,778,
%U 1002,897,1107,1030,1112,1205,1535,1343,1458,1093,1203,1588,1548,1822,1623,2162,2208,1577,1497,1948,2228,2473
%N Least positive integer m such that A254574(m) = n
%C Conjecture: (i) a(n) exists for any n > 0. Also, the main term of a(n) is n^2/2 as n tends to the infinity.
%C (ii) No term a(n) with n>2 is congruent to 1 or -1 modulo 5.
%C See also the comments in A254595 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A254617/b254617.txt">Table of n, a(n) for n = 1..200</a>
%H 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 a(4) = 20 since 20 is the first positive integer m with A254574(m) = 4. Note that 20 = 0*1/2 + 3*(3*3+1)/2 + 2*(3*2-1)/2 = 1*2/2 + 2*(3*2+1)/2 + 3*(3*3-1)/2 = 3*4/2 + 1*(3*1+1)/2 + 3*(3*3-1)/2 = 5*6/2 + 0*(3*0+1)/2 + 2*(3*2-1)/2.
%t TQ[n_]:=IntegerQ[Sqrt[8n+1]]
%t Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-y(3y+1)/2-z(3z-1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,(Sqrt[24m+1]-1)/6}, {z,0,(Sqrt[24(m-y(3y+1)/2)+1]+1)/6}];
%t If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]
%Y Cf. A000217, A000326, A005449, A254573, A254574, A254595.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Feb 03 2015