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%I #6 Nov 08 2019 22:30:23
%S 732722,23338590792,102091236,1,3,314,1,5,1,128,1,5,1,16,1,7,3,3,38,1,
%T 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,
%U 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
%N 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.
%C a(n) is the index in primes of A329224(n), or 0 if A329224(n) = 0.
%C In general, assuming the strong form of RH, if 0 < a, b < k, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod n, 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". This sequence gives the indices of the smallest primes q to violate the inequality Sum_{primes r <= q} Kronecker(q,p) <= 0, p = prime(n).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%e 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.
%o (PARI) 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))))))
%Y Cf. A306502, A306503. See A329224 for the actual primes.
%K nonn
%O 1,1
%A _Jianing Song_, Nov 08 2019
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