

A123489


a(n) = Sum_{k=0..p1} Kronecker(4k^3+1, p) where p is the nth prime of the form 3k+1.


1



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
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OFFSET

1,2


COMMENTS

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


REFERENCES

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.


LINKS

Table of n, a(n) for n=1..74.


EXAMPLE

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 3k+1 so a(5) = 11.


PROG

(PARI) {a(n)= local(p, c); if(n<1, 0, c=0; p=0; while(c<n, p++; if(isprime(p)& p%6==1, c++)); sum(x=0, p1, kronecker(4*x^3+1, p)) )}


CROSSREFS

A002338 is the unsigned version.
Sequence in context: A198998 A096437 A096458 * A002338 A324791 A226021
Adjacent sequences: A123486 A123487 A123488 * A123490 A123491 A123492


KEYWORD

sign


AUTHOR

Michael Somos, Sep 30 2006


STATUS

approved



