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%I #10 Dec 25 2015 12:50:42
%S 471,531,601,616,786,871,906,991,1056,1126,1156,1186,1281,1296,1341,
%T 1446,1486,1551,1576,1602,1641,1656,1771,1806,1836,1896,1906,1921,
%U 2044,2061,2146,2226,2265,2281,2316,2401,2421,2451,2486,2551,2646,2716,2746,2781
%N Numbers that are the sum of two distinct nonzero triangular numbers in exactly four ways.
%t r = 2781; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 4]
%t Module[{nn=80,trnos},trnos=Accumulate[Range[nn]];Select[PositionIndex[ Sort[ Counts[Total/@Subsets[trnos,{2}]]]][4],#<=Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* _Harvey P. Dale_, Dec 25 2015 *)
%Y Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways).
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Dec 02 2015
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