

A329241


Primes p such that Sum_{primes r <= q} Kronecker(r,p) <= 0 for all primes q <= p.


1



2, 3, 5, 13, 29, 43, 67, 163, 293, 677, 883, 907, 947, 1787, 1867, 2203, 2347, 2477, 2683, 3019, 3533, 3907, 4603, 5107, 5309, 5923, 6883, 7213, 7723, 7867, 8563, 9283, 9413, 9643, 10627, 10853, 11213, 12107, 13003, 13037, 13187, 14683, 14851, 15413, 15643, 15667, 15797
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OFFSET

1,1


COMMENTS

Primes p such that A329224(primepi(p)) > p (or equal to 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 primes p such that the smallest prime q to violate the inequality Sum_{primes r <= q} Kronecker(r,p) <= 0 is relatively large.
There are 141 primes in this sequence below 10^5 and 548 primes below 10^6.


LINKS

Jianing Song, Table of n, a(n) for n = 1..548 (all terms below 10^6)
Wikipedia, Chebyshev's bias


EXAMPLE

The smallest prime q such that Sum_{primes r <= q} Kronecker(r,2) = 1 > 0 is q = 11100143, so 2 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,3) = 1 > 0 is q = 608981813029, so 3 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,5) = 1 > 0 is q = 2082927221, so 5 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,13) = 1 > 0 is q = 2083, so 13 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,29) = 1 > 0 is q = 719, so 29 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,43) = 1 > 0 is q = 53, so 43 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,67) = 1 > 0 is q = 163, so 67 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,163) = 1 > 0 is q = 15073, so 163 is a term.
The smallest prime q such that Sum_{primes r <= q} Kronecker(r,293) = 1 > 0 is q = 349, so 293 is a term.


PROG

(PARI) isA329241(p) = if(isprime(p), my(i=0); forprime(q=2, p, i+=kronecker(q, p); if(i>0, return(0))); return(1), 0)


CROSSREFS

Cf. A329224.
Sequence in context: A099962 A173718 A281316 * A177502 A072536 A087592
Adjacent sequences: A329238 A329239 A329240 * A329242 A329243 A329244


KEYWORD

nonn


AUTHOR

Jianing Song, Nov 08 2019


STATUS

approved



