%I
%S 2,7,13,17,19,31,37,43,53,61,67,71,79,89,97,103,107,127,139,151,157,
%T 163,179,193,197,211,223,229,233,241,251,269,271,277,283,313,331,337,
%U 349,359,367,373,379,397,409,421,431,433,439,449,457,463,467,499,503,521
%N Primes representable as the sum of two rational cubes.
%C The prime elements of A159843, i.e., the intersection of A159843 and A000040.
%C Also, the prime elements of A020898.
%D H. Cohen, Number Theory. I, Tools and Diophantine Equations, SpringerVerlag, 2007, p. 378.
%F Under the Birch and SwinnertonDyer conjecture, these primes consist of:
%F (i) p = 2;
%F (ii) p == 4, 7, or 8 (mod 9);
%F (iii) p == 1 (mod 9) and p divides A206309(p1), i.e., VillegasZagier polynomial A166243((p1)/3) evaluated at x=0.
%t (* To speed up computation, a few terms are precomputed *) nmax = 521; xmax = 360; preComputed = {127, 271, 379}; solQ[p_] := Do[ If[ IntegerQ[z = Root[x^3  y^3 + p*#^3 & , 1]], Print[p, {x, y, z}]; Return[True]], {x, 2, xmax}, {y, x, xmax}]; A166246 = Union[ preComputed, Select[ Prime[ Range[ PrimePi[nmax]]], Mod[#, 9] == 4  Mod[#, 9] == 7  Mod[#, 9] == 8  solQ[#] === True & ]](* _JeanFrançois Alcover_, Apr 04 2012, after given formula *)
%Y Cf. A166243, A166244, A159843.
%K nonn
%O 1,1
%A _Max Alekseyev_, Oct 10 2009
