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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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)

# Function 'is_fundamental' is defined in A003658.

def KroneckerSum():

    yield 2

    ind = 0

    while true:

        ind += 1

        while not is_fundamental(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

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Last modified January 26 20:11 EST 2020. Contains 331288 sequences. (Running on oeis4.)