A002339 Positive y such that p = (x^2 + 27y^2)/4 where p is the n-th prime of the form 6k+1. (Formerly M0058 N0043)

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, 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, 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

Offset

1,4

Comments

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

References

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.

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. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Ruperto Corso, Table of n, a(n) for n = 1..1000

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Example

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

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A002338, A002476, A123489.

Sequence in context: A104345 A244516 A363264 * A123243 A037193 A291615

Adjacent sequences: A002336 A002337 A002338 * A002340 A002341 A002342

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Corrected and extended by Ruperto Corso, Dec 14 2011

Name clarified by Michael Somos, Jul 10 2022

Status

approved