A002338 -id:A002338 - cp's OEIS frontend

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

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

N. J. A. Sloane

nonn

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

			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

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).

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.

Cf. A002338, A002476, A123489.

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 *)

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 */

Corrected and extended by Ruperto Corso, Dec 14 2011

Name clarified by Michael Somos, Jul 10 2022

A123489 a(n) = Sum_{k=0..p-1} Kronecker(4k^3+1, p) where p is the n-th prime of the form 6k+1.

Original entry on oeis.org

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, -26, -32, 16, -35, 1, -5, 37, -35, 13, -29, 34, 31, 19, -2, 28, 10, -23, 25, -32, 43, -29, 1, 31, -11, -26, 49, -47, -17, 43, 40, 49, 37, -8, -53, -44, -50, 16, -41, -29, 49, 31, -56, -5, 7, -35, 13, -59, -47

Offset: 1

Michael Somos, Sep 30 2006

sign

Given a prime p == 1 (mod 6), the sum x is the unique solution to 4*p = x^2 + 27*y^2 where x == 1 (mod 3) and y is a positive integer.

Given a prime p == 1 (mod 6), the number of solutions to u^3 + v^3 == 1 (mod p) is p - 2 + x. - Michael Somos, Jul 11 2022

			If p = 37, then 4*37 = (-11)^2 + 27*(1)^2 where -11 = Sum_{k=0..36} Kronecker(4k^3+1, 37) and 37 is the 5th prime of the form 6k+1 so a(5) = -11.
If p = 13, then 4*p = x^2 + 26*y^2 where x = -5, y = 1. The solutions to u^3 + v^3 == 1 (mod p) is {(0,1), (1,0), (0,3), (3,0), (0,9), (9,0)} with cardinality = 6 = 13 - 2 + (-5). - _Michael Somos_, Jul 11 2022

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 55.

Cf. A002338 is the unsigned version, A002339 is the y.

Cf. A002476 (primes of the form 6k+1).

{a(n) = my(p, c); if(n<1, 0, c=0; p=0; while(c

cp's OEIS Frontend

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

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