A329225 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.

732722, 23338590792, 102091236, 1, 3, 314, 1, 5, 1, 128, 1, 5, 1, 16, 1, 7, 3, 3, 38, 1, 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, 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

Offset

1,1

Comments

a(n) is the index in primes of A329224(n), or 0 if A329224(n) = 0.

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).

Links

Table of n, a(n) for n=1..77.

Wikipedia, Chebyshev's bias

Example

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.

Prog

(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))))))

Crossrefs

Cf. A306502, A306503. See A329224 for the actual primes.

Sequence in context: A156867 A156868 A238297 * A236901 A107447 A184504

Adjacent sequences: A329222 A329223 A329224 * A329226 A329227 A329228

Keyword

nonn

Author

Jianing Song, Nov 08 2019

Status

approved