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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.

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.

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

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

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

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

Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

(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)))))

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

11100143, 608981813029, 2082927221, 2

1,1

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

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nonn

Jianing Song, Nov 08 2019

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allocated for Jianing Song

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