A392284 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, 3, 608981813029, 26861, 7, 5, 2, 3, 2, 11, 5, 608981813017, 19, 3, 2, 26861, 2, 643, 11, 3, 11, 31, 2, 5, 2, 3, 608981813029, 48731, 5, 13, 2, 3, 2, 7, 11, 5, 199, 3, 2, 11, 2, 29, 53, 3, 109, 41, 2, 608981813017, 2, 3, 13, 17, 23, 5, 2, 3, 2, 1019, 5, 263, 11, 3, 2, 26861

Offset

1,1

Comments

Write P = 608981813017, then Sum_{q prime<=p} kronecker(-3,q) <= -1 for 2 <= p < P (see A096449), and is equal to 0 for p = P.

For n = 3^e*m^2, e odd, gcd(m,3) = 1:

(a) if m has no prime factor < P, then Sum_{q prime<=p} kronecker(-n,q) = Sum_{q prime<=p} kronecker(-3,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(-3,q) < 0 for primes p < lpf(m), where lpf = least prime factor;

- Sum_{q prime<=p} kronecker(-n,q) = Sum_{q prime<=p} kronecker(-3,q) - Sum_{q|m, q<P} kronecker(-3,q) for primes r <= p < P.

As a result, if Sum_{q|m, q<P} kronecker(-3,q) >= -1, then either lpf(m) < a(n) < r, or a(n) >= P.

Similarly, write P = 24996190781. For n = 1605^e*m^2, e odd, gcd(m,1605) = 1, if Sum_{q|m, q<P} kronecker(-1605,q) >= -9, then we have either lpf(m) < a(n) < r (where r is the largest prime factor < P of m) or a(n) >= P, unless the prime factors < P of m are exactly

- {2,7} (then a(n) = 11);

- {2,13,17,19,23,29} (then a(n) = 31);

- {7,13,17,19,23,29} (then a(n) = 31);

- {2,7,11,13,17,19,23,29} (then a(n) = 31).

See my link below for more details.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Jianing Song, Finding a(1605^e*m^2), e odd, gcd(m,1605) = 1

Wikipedia, Chebyshev's bias

Formula

a(A003657(n)) = A306500(n).

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

a(n) = 3 iff n == 2 (mod 6).

Prog

(PARI) a(n) = {
if(issquare(n/3), my(P=608981813017, temp=n, f, s=0, maxp=0); while(temp%3==0, temp/=3); f=factor(temp); for(i=1, #f~, if(f[i, 1]<P, s+=kronecker(-3, f[i, 1]); maxp=f[i, 1], break())); if(s>=-1, my(i=0); forprime(p=2, maxp, i+=kronecker(-n, p); if(i>0, return(p))); i=-1-s; forprime(p=P, oo, i+=kronecker(-n, p); if(i>0, return(p))); ); ); \\ the case n = 3^e*m^2, Sum_{q|m, q<P} kronecker(-3, q) >= -1
my(i=0); forprime(p=2, oo, i+=kronecker(-n, p); if(i>0, return(p)))}

Crossrefs

Cf. A326615.

A306500 is the subsequence for negated fundamental discrimiants of imaginary quadratic fields (A003657).

Sequence in context: A176649 A379989 A216979 * A334822 A082871 A321406

Adjacent sequences: A392281 A392282 A392283 * A392285 A392286 A392287

Keyword

nonn

Author

Jianing Song, Jan 06 2026

Status

approved