cp's OEIS Frontend

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.

%I A329224 #13 Jun 26 2025 08:09:28
%S A329224 11100143,608981813029,2082927221,2,5,2083,2,11,2,719,2,11,2,53,2,17,
%T A329224 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,
%U A329224 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
%N 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.
%C A329224 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]
%C A329224 This sequence gives the smallest primes q to violate the inequality Sum_{primes r <= q} Kronecker(r,p) <= 0, for p = prime(n).
%H A329224 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>
%H A329224 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%e A329224 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.
%o A329224 (PARI) 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)))))
%Y A329224 Cf. A007350, A306499, A306500, A329225 (indices of these primes).
%K A329224 nonn
%O A329224 1,1
%A A329224 _Jianing Song_, Nov 08 2019