|
%I M0058 N0043 #34 Jul 23 2022 09:45:03
%S 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 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 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 Positive y such that p = (x^2 + 27y^2)/4 where p is the n-th prime of the form 6k+1.
%C 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 A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
%D 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 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ruperto Corso, <a href="/A002339/b002339.txt">Table of n, a(n) for n = 1..1000</a>
%H A. J. C. Cunningham, <a href="/A002330/a002330.pdf">Quadratic Partitions</a>, Hodgson, London, 1904 [Annotated scans of selected pages]
%H 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 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 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 (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 Cf. A002338, A002476, A123489.
%K nonn
%O 1,4
%A _N. J. A. Sloane_
%E Corrected and extended by _Ruperto Corso_, Dec 14 2011
%E Name clarified by _Michael Somos_, Jul 10 2022
|
