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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.

%I A306499 #36 Mar 24 2020 17:09:37
%S A306499 2,2082927221,11100143,61463,2083,2,1217,5,3,719,2,11,3,2,7,17,11,2,7,
%T A306499 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,
%U A306499 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
%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.
%C A306499 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.
%C A306499 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.
%C A306499 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).
%H A306499 Jianing Song, <a href="/A306499/b306499.txt">Table of n, a(n) for n = 1..3044</a>
%H A306499 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%F A306499 a(n) = 2 if A003658(n) == 1 (mod 8);
%F A306499 a(n) = 3 if A003658(n) == 28, 40 (mod 48);
%F A306499 a(n) = 5 if A003658(n) == 24, 61, 109, 156, 181, 204, 216, 229 (mod 240).
%e A306499 Let D = A003658(16) = 53, j(k) = Sum_{primes q <= prime(k)} Kronecker(D,q).
%e A306499 For k = 1, Kronecker(53,2) = -1, so j(1) = -1;
%e A306499 For k = 2, Kronecker(53,3) = -1, so j(2) = -2;
%e A306499 For k = 3, Kronecker(53,5) = -1, so j(3) = -3;
%e A306499 For k = 4, Kronecker(53,7) = +1, so j(4) = -2;
%e A306499 For k = 5, Kronecker(53,11) = +1, so j(5) = -1;
%e A306499 For k = 6, Kronecker(53,13) = +1, so j(6) = 0;
%e A306499 For k = 7, Kronecker(53,17) = +1, so j(7) = 1.
%e A306499 The first time for j > 0 occurs at the prime 17, so a(16) = 17.
%o A306499 (PARI) b(n) = my(i=0); forprime(p=2,oo,i+=kronecker(n,p); if(i>0, return(p)))
%o A306499 for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))
%o A306499 (Sage)
%o A306499 def KroneckerSum():
%o A306499     yield 2
%o A306499     ind = 0
%o A306499     while True:
%o A306499         ind += 1
%o A306499         while not is_fundamental_discriminant(ind):
%o A306499             ind += 1
%o A306499         s, p = 0, 1
%o A306499         while s < 1:
%o A306499             p = p.next_prime()
%o A306499             s += kronecker(ind, p)
%o A306499         yield p
%o A306499 A306499 = KroneckerSum()
%o A306499 print([next(A306499) for _ in range(71)]) # _Peter Luschny_, Feb 26 2019
%Y A306499 Cf. A003658, A306500 (the negative discriminants case).
%Y A306499 The indices of primes are given in A306502.
%K A306499 nonn
%O A306499 1,1
%A A306499 _Jianing Song_, Feb 19 2019