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Search: id:A123489
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| A123489 |
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a(n) = Sum_{k=0..p-1} kronecker(4k^3+1, p) where p is the n-th prime of the form 3k+1. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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.
A002838 is the unsigned version.
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REFERENCES
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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.
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EXAMPLE
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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.
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PROGRAM
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(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, p-1, kronecker(4*x^3+1, p)) )}
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CROSSREFS
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Adjacent sequences: A123486 A123487 A123488 this_sequence A123490 A123491 A123492
Sequence in context: A096437 A096458 A002338 this_sequence A165242 A104542 A161376
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Sep 30 2006
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