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

1,1

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).

Links

Jianing Song, Table of n, a(n) for n = 1..3044

Wikipedia, Chebyshev's bias

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).

Example

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.

Prog

(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

Crossrefs

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

The indices of primes are given in A306502.

Sequence in context: A249482 A214600 A325175 * A051241 A094486 A109741

Adjacent sequences: A306496 A306497 A306498 * A306500 A306501 A306502

Keyword

nonn

Author

Jianing Song, Feb 19 2019

Status

approved