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A002338 x such that p = (x^2 + 27*y^2)/4, where p is the n-th prime of the form 3k+1.

%I A002338 M3754 N1531 #31 Sep 06 2018 08:23:44
%S A002338 1,5,7,4,11,8,1,5,7,17,19,13,2,20,23,19,14,25,7,23,11,13,28,22,17,29,
%T A002338 26,32,16,35,1,5,37,35,13,29,34,31,19,2,28,10,23,25,32,43,29,1,31,11,
%U A002338 26,49,47,17,43,40,49,37,8,53,44,50,16,41,29,49,31,56,5,7,35,13,59,47,19,52,61,41,61,10,43,14,53,59,64,65,62,55,22,65,35,67,7
%N A002338 x such that p = (x^2 + 27*y^2)/4, where p is the n-th prime of the form 3k+1.
%C A002338 A123489 is a signed version. - _Michael Somos_, Aug 27 2012
%D A002338 A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
%D A002338 B. Engquist and Wilfried Schmid, Mathematics Unlimited - 2001 and Beyond, Chapter on Error-correcting codes and curves over finite fields, see pp. 1118-1119. [From Neven Juric, Oct 16 2008.]
%D A002338 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%D A002338 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002338 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002338 Ruperto Corso, <a href="/A002338/b002338.txt">Table of n, a(n) for n = 1..1000</a>
%H A002338 A. J. C. Cunningham, <a href="/A002330/a002330.pdf">Quadratic Partitions</a>, Hodgson, London, 1904 [Annotated scans of selected pages]
%H A002338 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, (arXiv:math.NT/0701251).
%t A002338 Reap[For[p = 2, p<2000, p = NextPrime[p], For[x = 1, x <= Floor[2Sqrt[p]], x++, px = 4p - x^2; If[Mod[px, 27] == 0, If[IntegerQ[Sqrt[px/27]], Sow[x] ]]]]][[2, 1]] (* _Jean-François Alcover_, Sep 06 2018, after _Ruperto Corso_ *)
%o A002338 (PARI) forprime(p=2,10000,for(x=1,floor(2*sqrt(p)),px=4*p-x^2;if(px%27==0,if(issquare(px/27,&y),print1(x","))))) /* _Ruperto Corso_, Dec 14 2011 */
%Y A002338 Cf. A002339, A123489.
%K A002338 nonn
%O A002338 1,2
%A A002338 _N. J. A. Sloane_
%E A002338 Corrected and extended by _Ruperto Corso_, Dec 14 2011