This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002339 M0058 N0043 #34 Jul 23 2022 09:45:03 %S A002339 1,1,1,2,1,2,3,3,3,1,1,3,4,2,1,3,4,1,5,3,5,5,2,4,5,3,4,2,6,1,7,7,1,3, %T A002339 7,5,4,5,7,8,6,8,7,7,6,3,7,9,7,9,8,1,3,9,5,6,3,7,10,1,6,4,10,7,9,5,9, %U A002339 2,11,11,9,11,1,7,11,6,1,9,3,12,9,12,7,5,2,1,4,7,12,3,11,1,13,13,7,13,13,11,9,11,5,13,9,3,14,13,6,14,5,13,7,10,2,13,1,15,3,15 %N A002339 Positive y such that p = (x^2 + 27y^2)/4 where p is the n-th prime of the form 6k+1. %C A002339 Given a prime p = 6k+1, then there exists a unique pair of integers (x, y) such that 4p = x^2 + 27y^2, x == 1 (mod 3), and y>0. - _Michael Somos_, Jul 10 2022 %D A002339 A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1. %D A002339 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 A002339 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55. %D A002339 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002339 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A002339 Ruperto Corso, <a href="/A002339/b002339.txt">Table of n, a(n) for n = 1..1000</a> %H A002339 A. J. C. Cunningham, <a href="/A002330/a002330.pdf">Quadratic Partitions</a>, Hodgson, London, 1904 [Annotated scans of selected pages] %H A002339 Steven R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007. %e A002339 The 7th prime of the form 6k+1 (A002476) is 61 and 4*61 = 244 = 1^2 + 27*3^2 gives a(7) = 3. The 8th prime of the form 6k+1 is 67 and 4*67 = 268 = (-5)^2 + 27*3^2 gives a(8) = 3. - _Michael Somos_, Jul 10 2022 %t A002339 Reap[For[p = 2, p<2000, p = NextPrime[p], For[x = 1, x <= Floor[2*Sqrt[p]], x++, px = 4*p - x^2; If[Mod[px, 27] == 0, If[IntegerQ[y = Sqrt[px/27]], Sow[y]]]]]][[2, 1]] (* _Jean-François Alcover_, Sep 06 2018, after _Ruperto Corso_ *) %o A002339 (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(y","))))) /* _Ruperto Corso_, Dec 14 2011 */ %Y A002339 Cf. A002338, A002476, A123489. %K A002339 nonn %O A002339 1,4 %A A002339 _N. J. A. Sloane_ %E A002339 Corrected and extended by _Ruperto Corso_, Dec 14 2011 %E A002339 Name clarified by _Michael Somos_, Jul 10 2022