A326615 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(n,q) > 0, or 0 if no such prime exists.

2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3

Offset

1,1

Comments

From Jianing Song, Jan 06 2026: (Start)

Write P = 2082927019, then Sum_{q prime<=p} kronecker(5,q) <= -2 for 2 < p < P, and is equal to -1 for p = P.

For n = 5^e*m^2, e odd, gcd(m,5) = 1:

(a) if m has no prime factor < P, then Sum_{q prime<=p} kronecker(n,q) = Sum_{q prime<=p} kronecker(5,q) < 0 for primes p < P, so a(n) > P.

(b) if m has prime factors < P, let r be the largest such factor. Then

- Sum_{q prime<=p} kronecker(n,q) = Sum_{q prime<=p} kronecker(5,q) < 0 for primes p < lpf(m), where lpf = least prime factor;

- Sum_{q prime<=p} kronecker(n,q) = Sum_{q prime<=p} kronecker(5,q) - Sum_{q|m, q<P} kronecker(5,q) for primes r <= p < P.

As a result, if Sum_{q|m, q<P} kronecker(5,q) >= -2, then either lpf(m) < a(n) < r, or a(n) >= P.

(End)

Links

Eric Chen, Table of n, a(n) for n = 1..4096

Wikipedia, Chebyshev's bias

Formula

a(A003658(n)) = A306499(n).

a(n) = 2 iff n == 1 or 7 mod 8 (see A047522).

a(n) = 3 iff n == 4 mod 6 (see A016957).

Prog

(PARI) a(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p))) \\ after Jianing Song in A306499
(PARI) a(n) = {
if(issquare(n/5), my(P=2082927019, temp=n, f, s=0, maxp=0); while(temp%5==0, temp/=5); f=factor(temp); for(i=1, #f~, if(f[i, 1]<P, s+=kronecker(5, f[i, 1]); maxp=f[i, 1], break())); if(s>=-2, my(i=0); forprime(p=2, maxp, i+=kronecker(n, p); if(i>0, return(p))); i=-2-s; forprime(p=P, oo, i+=kronecker(n, p); if(i>0, return(p))); ); ); \\ the case n = 5^e*m^2, Sum_{q|m, q<P} kronecker(5, q) >= -2
my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p)))} \\ Jianing Song, Jan 06 2026; Should be a bit faster to generate a(n) for n = 5^e*m^2

Crossrefs

Cf. A392284.

A306499 is the subsequence for fundamental discrimiants of real quadratic fields (A003658).

Sequence in context: A324440 A121390 A170997 * A259407 A157992 A322145

Adjacent sequences: A326612 A326613 A326614 * A326616 A326617 A326618

Keyword

nonn

Author

Eric Chen, Jul 15 2019

Status

approved