A392284 - OEIS

%I #16 Jan 07 2026 08:37:31

%S 2,3,608981813029,26861,7,5,2,3,2,11,5,608981813017,19,3,2,26861,2,

%T 643,11,3,11,31,2,5,2,3,608981813029,48731,5,13,2,3,2,7,11,5,199,3,2,

%U 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

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

%C 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.

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

%C (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.

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

%C - 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;

%C - 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.

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

%C 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

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

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

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

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

%C See my link below for more details.

%H Jianing Song, <a href="/A392284/b392284.txt">Table of n, a(n) for n = 1..10000</a>

%H Jianing Song, <a href="/A392284/a392284.pdf">Finding a(1605^e*m^2), e odd, gcd(m,1605) = 1</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

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

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

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

%o (PARI) a(n) = {

%o 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

%o my(i=0); forprime(p=2, oo, i+=kronecker(-n, p); if(i>0, return(p)))}

%Y Cf. A326615.

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

%K nonn

%O 1,1

%A _Jianing Song_, Jan 06 2026