A306500 -id:A306500 - cp's OEIS frontend

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

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)), ", ")))

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.

Original entry on oeis.org

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).

A329224 a(n) is the smallest prime q such that Sum_{primes r <= q} Kronecker(r,prime(n)) > 0 (or equivalently, Sum_{primes r <= q} Kronecker(r,prime(n)) = 1), or 0 if no such prime exists.

Original entry on oeis.org

11100143, 608981813029, 2082927221, 2, 5, 2083, 2, 11, 2, 719, 2, 11, 2, 53, 2, 17, 5, 5, 163, 2, 2, 2, 11, 2, 2, 23, 2, 23, 5, 2, 2, 5, 2, 11, 31, 2, 17, 15073, 2, 47, 5, 5, 2, 2, 47, 2, 59, 2, 11, 5, 2, 2, 2, 5, 2, 2, 47, 2, 23, 2, 97, 349, 103, 2, 2, 67, 149, 2, 67

Offset: 1

Jianing Song, Nov 08 2019

nonn

In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not, where Pi(k,b)(n) is the number of primes <= n that are congruent to b modulo k. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Jun 26 2025]

This sequence gives the smallest primes q to violate the inequality Sum_{primes r <= q} Kronecker(r,p) <= 0, for p = prime(n).

			For prime(6) = 13, q = 2083 is the first case such that Sum_{primes r <= q} Kronecker(r,13) = 1 > 0, so a(6) = 2083.

Eric Weisstein's World of Mathematics, Kronecker Symbol
Wikipedia, Chebyshev's bias

Cf. A007350, A306499, A306500, A329225 (indices of these primes).

a(n) = if(n==2, 608981813029, if(n==3, 2082927221, my(p=prime(n), i=0); forprime(q=2, oo, i+=kronecker(q, p); if(i>0, return(q)))))

A326615 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(n,q) > 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3

Offset: 1

Richard N. Smith, Jul 15 2019

nonn

Richard N. Smith, Table of n, a(n) for n = 1..4096

Cf. A306499, A306500.

a(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p))) \\ after Jianing Song in A306499

a(A003658(n)) = A306499(n).
a(n) = 2 iff n == 1 or 7 mod 8 (see A047522).
a(n) = 3 iff n == 4 mod 6 (see A016957).

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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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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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A329224 a(n) is the smallest prime q such that Sum_{primes r <= q} Kronecker(r,prime(n)) > 0 (or equivalently, Sum_{primes r <= q} Kronecker(r,prime(n)) = 1), or 0 if no such prime exists.

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PARI

A326615 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(n,q) > 0, or 0 if no such prime exists.

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