A329241 Primes p such that Sum_{primes r <= q} Kronecker(r,p) <= 0 for all primes q <= p.

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

Offset: 1

Author

Jianing Song, Nov 08 2019

Keywords

• nonn

Comments

Primes p such that A329224(primepi(p)) > p (or equal to 0).

So, in terms of the above comparison, 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.

See also the comments and references in A329224, which is the main entry for this set of sequences.

There are 141 primes in this sequence below 10^5 and 548 primes below 10^6.

Examples

			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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..548 (all terms below 10^6)

Crossrefs

Cf. A329224.

Programs

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)

Extensions

Edited by Peter Munn, Jun 27 2025