A306502 -id:A306502 - cp's OEIS frontend

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

Jianing Song, Feb 19 2019

nonn

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).

			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.

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

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

The indices of primes are given in A306502.

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), ", ")))

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

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).

A306503 a(n) is the index in primes of A306500(n), or 0 if A306500(n) = 0.

Original entry on oeis.org

23338590792, 2946, 1, 2, 3, 1, 5, 2, 1, 3, 1, 5, 1, 5, 16, 1, 6, 7, 1, 2, 3, 38, 2, 1, 1, 5, 3, 1, 11, 11, 1, 1, 2, 9, 1, 13, 2, 1, 6, 15, 1, 3, 8, 4, 5, 1, 43, 1, 2, 8, 1, 1760, 2, 1, 10, 3, 1, 13, 29, 1, 5, 1, 11, 17, 2, 1, 8, 1, 5, 16, 1, 171, 33, 1, 4, 1, 2, 3

Offset: 1

Jianing Song, Feb 19 2019

nonn

a(n) is the smallest integer k such that Sum_{i=1..k} Kronecker(-A003657(n),prime(i)) > 0, or 0 if no such k exists.

See A306500 for the actual primes.

			See A306500 for the example that shows a(18) = 7.

Cf. A003657, A306500, A306502.

b(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p)))
print1(23338590792, ", "); for(n=4, 300, if(isfundamental(-n), print1(primepi(b(-n)), ", ")))

A329225 a(n) is the smallest number k such that Sum_{i=1..k} Kronecker(prime(i),prime(n)) > 0 (or equivalently, Sum_{i=1..k} Kronecker(prime(i),prime(n)) = 1), or 0 if no such k exists.

Original entry on oeis.org

732722, 23338590792, 102091236, 1, 3, 314, 1, 5, 1, 128, 1, 5, 1, 16, 1, 7, 3, 3, 38, 1, 1, 1, 5, 1, 1, 9, 1, 9, 3, 1, 1, 3, 1, 5, 11, 1, 7, 1760, 1, 15, 3, 3, 1, 1, 15, 1, 17, 1, 5, 3, 1, 1, 1, 3, 1, 1, 15, 1, 9, 1, 25, 70, 27, 1, 1, 19, 35, 1, 19, 3, 1, 1, 1, 7, 41, 1, 5

Offset: 1

Jianing Song, Nov 08 2019

nonn

a(n) is the index in primes of A329224(n), or 0 if A329224(n) = 0.

For further information see A329224, which is the main entry for these sequences.

			For prime(10) = 29, k = 128 is the first case such that Sum_{i=1..k} Kronecker(prime(i),29) = 1 > 0, so a(10) = 128.

Cf. A306502, A306503. See A329224 for the actual primes.

a(n) = if(n==2, 23338590792, if(n==3, 102091236, my(p=prime(n), i=0); forprime(q=2, oo, i+=kronecker(q, p); if(i>0, return(primepi(q))))))

Edited by Peter Munn, Jun 26 2025

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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.

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A306503 a(n) is the index in primes of A306500(n), or 0 if A306500(n) = 0.

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A329225 a(n) is the smallest number k such that Sum_{i=1..k} Kronecker(prime(i),prime(n)) > 0 (or equivalently, Sum_{i=1..k} Kronecker(prime(i),prime(n)) = 1), or 0 if no such k exists.

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