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A329224
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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(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)))))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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