

A002339


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


3



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
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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 Errorcorrecting codes and curves over finite fields, see pp. 11181119. [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

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


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]] (* JeanFrançois Alcover, Sep 06 2018, after Ruperto Corso *)


PROG

(PARI) forprime(p=2, 10000, for(x=1, floor(2*sqrt(p)), px=4*px^2; if(px%27==0, if(issquare(px/27, &y), print1(y", "))))) /* Ruperto Corso, Dec 14 2011 */


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



