%I
%S 0,1,3,4,7,9,10,19,24,25,34,37,39,42,49,54,55,72,73,78,85,87,93,94,
%T 102,108,109,118,138,142,147,157,160,165,168,175,192,195,202,210,214,
%U 220,228,232,243,247,249,250,252,253,258,267,273,274,279,289,297,312,333
%N Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.
%C A115104 is a subsequence. Terms such that 4*n^3 + 1 is not prime are 24, 337, 457, 750, 840, 1015, ...
%e 3 is a term because 3^3 = 27 = 6 + 21.
%t Select[Range@ 333, Length[PowersRepresentations[4 #^3 + 1, 2, 2]] == 1 &] (* after _Ant King_ at A052343, or *)
%t nn = 20; t = (#^2 + #)/2 & /@ Range[0, nn^3]; Select[Range[0, nn], Function[n, Count[Transpose@ {#, n^3  #} &@ Range[0, Floor[n^3/2]], k_ /; Times @@ Boole@ Map[MemberQ[t, #] &, k] == 1] == 1]] (* _Michael De Vlieger_, Jul 07 2016 *)
%o (PARI) a052343(n) = sum(i=0, (sqrtint(4*n + 1)  1)\2, issquare(n  i  i^2));
%o lista(nn) = for(n=0, nn, if(a052343(n^3) == 1, print1(n, ", ")));
%Y Cf. A000217, A000578, A020756, A052343, A115104, A119345.
%K nonn
%O 1,3
%A _Altug Alkan_, Jul 07 2016
