A329224 -id:A329224 - cp's OEIS frontend

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

Jianing Song, Nov 08 2019

nonn

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

For further information see A329224, which is the main entry for these sequences.

			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.

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

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))))))

Edited by Peter Munn, Jun 26 2025

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

Original entry on oeis.org

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

Jianing Song, Nov 08 2019

nonn

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.

			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.

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

Cf. A329224.

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

Edited by Peter Munn, Jun 27 2025

cp's OEIS Frontend

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.

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