A306499 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(A003658(n),q) > 0, or 0 if no such prime exists.

2, 2082927221, 11100143, 61463, 2083, 2, 1217, 5, 3, 719, 2, 11, 3, 2, 7, 17, 11, 2, 7, 5, 2, 13, 2, 3, 23, 7, 3, 2, 13, 19, 2, 23, 17, 2, 5, 2, 7, 3, 2, 13, 3, 2, 19, 7, 2, 31, 31, 5, 17, 2, 13, 13, 3, 47, 2, 5, 3, 2, 37, 2, 47, 2, 5, 7, 2, 43, 2, 3, 11, 5, 3, 2, 29

Offset: 1

Author

Jianing Song, Feb 19 2019

Keywords

• nonn

Comments

Let D be a fundamental discriminant (only the case where D is fundamental is considered because {Kronecker(D,k)} forms a primitive real Dirichlet character with period |D| if and only if D is fundamental), it seems that Sum_{primes q <= p} Kronecker(D,p) <= 0 occurs much more often than its opposite does. This can be seen as a variation of the well-known "Chebyshev's bias". Sequence gives the least prime that violates the inequality when D runs through all positive discriminants.

For any D, the primes p such that Kronecker(D,p) = 1 has asymptotic density 1/2 in all the primes, so a(n) should be > 0 for all n.

Actually, for most n, a(n) is relatively small compared with A003658(n). There are only 52 n's in [1, 3044] (there are 3044 terms in A003658 below 10000) such that a(n) > A003658(n). The largest terms among the 52 corresponding terms are a(2) = 2082927221 (with A003658(2) = 5), a(2193) = 718010179 (with A003658(2193) = 7213) and a(3) = 11100143 (with A003658(3) = 8).

Examples

			Let D = A003658(16) = 53, j(k) = Sum_{primes q <= prime(k)} Kronecker(D,q).
For k = 1, Kronecker(53,2) = -1, so j(1) = -1;
For k = 2, Kronecker(53,3) = -1, so j(2) = -2;
For k = 3, Kronecker(53,5) = -1, so j(3) = -3;
For k = 4, Kronecker(53,7) = +1, so j(4) = -2;
For k = 5, Kronecker(53,11) = +1, so j(5) = -1;
For k = 6, Kronecker(53,13) = +1, so j(6) = 0;
For k = 7, Kronecker(53,17) = +1, so j(7) = 1.
The first time for j > 0 occurs at the prime 17, so a(16) = 17.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..3044
• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A003658, A306500 (the negative discriminants case).

The indices of primes are given in A306502.

Programs

PARI

b(n) = my(i=0); forprime(p=2,oo,i+=kronecker(n,p); if(i>0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

Sage

def KroneckerSum():
    yield 2
    ind = 0
    while True:
        ind += 1
        while not is_fundamental_discriminant(ind):
            ind += 1
        s, p = 0, 1
        while s < 1:
            p = p.next_prime()
            s += kronecker(ind, p)
        yield p
A306499 = KroneckerSum()
print([next(A306499) for  in range(71)]) # _Peter Luschny, Feb 26 2019

Formula

a(n) = 2 if A003658(n) == 1 (mod 8);

a(n) = 3 if A003658(n) == 28, 40 (mod 48);

a(n) = 5 if A003658(n) == 24, 61, 109, 156, 181, 204, 216, 229 (mod 240).