A329225 -id:A329225 - 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.

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

Offset: 1

Jianing Song, Nov 08 2019

nonn

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]

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

			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.

Eric Weisstein's World of Mathematics, Kronecker Symbol
Wikipedia, Chebyshev's bias

Cf. A007350, A306499, A306500, A329225 (indices of these primes).

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)))))

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI