%I A123489
%S A123489 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,
%T A123489 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,
%U A123489 47,17,43,40,49,37,8,53,44,50,16,41,29,49,31,56,5,7,35,13,59,47
%V A123489 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,
%W A123489 -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,
%X A123489 -17,43,40,49,37,-8,-53,-44,-50,16,-41,-29,49,31,-56,-5,7,-35,13,-59,-47
%N A123489 a(n) = Sum_{k=0..p-1} kronecker(4k^3+1, p) where p is the n-th prime 
               of the form 3k+1.
%C A123489 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.
%C A123489 A002838 is the unsigned version.
%D A123489 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.
%e A123489 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.
%o A123489 (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)) )}
%Y A123489 Sequence in context: A096437 A096458 A002338 this_sequence A165242 A104542 
               A161376
%Y A123489 Adjacent sequences: A123486 A123487 A123488 this_sequence A123490 A123491 
               A123492
%K A123489 sign
%O A123489 1,2
%A A123489 Michael Somos, Sep 30 2006