%I
%S 1,2,6,7,8,9,12,13,15,16,17,19,20,22,26,27,28,30,31,33,34,35,37,42,43,
%T 48,49,50,51,53,54,56,58,61,62,63,64,65,67,68,69,70,71,72,75,78,79,84,
%U 85,86,87,89,90,91,92,94,96,97,98,103,104,105,106,107,110,114,115,117
%N Sums of two rational cubes.
%C Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.
%C The prime elements are listed in A166246.  _Max Alekseyev_, Oct 10 2009
%D H. Cohen, Number Theory. I, Tools and Diophantine Equations, SpringerVerlag, 2007, p. 379.
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A cubefree integer c>2 is in this sequence iff the elliptic curve y^2=x^3+16*c^2 has positive rank.  _Max Alekseyev_, Oct 10 2009
%t (* A naive program with a few precomputed terms *) nmax = 117; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Join[{1}, Reap[ Do[n = CubeFreePart[x*y*(x + y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union]; A159843 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* _JeanFrançois Alcover_, Apr 03 2012 *)
%Y Cf. A020894, A020895, A020897, A020898.
%Y Complement of A185345.
%K nice,nonn
%O 1,2
%A _Steven Finch_, Apr 23 2009
