A321859 - OEIS

%I #17 Nov 19 2023 10:23:55

%S 0,-1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,2,2,3,3,3,3,2,2,2,2,2,2,1,1,2,2,2,2,

%T 2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,1,1,1,1,1,1,2,2,3,3,3,3,3,3,2,2,

%U 2,2,1,1,2,2,2,2,2,2,1,1,1,1,2,2,2,2,2

%N Number of primes congruent to 3, 5, 6 modulo 7 and <= n minus number of primes congruent to 1, 2, 4 modulo 7 and <= n.

%C a(n) is the number of primes <= n that are quadratic nonresidues modulo 7 minus the number of primes <= n that are quadratic residues modulo 7.

%C The first 10000 terms (except for a(2)) are nonnegative. a(p) = 0 for primes p = 3, 11, 211, 3371, 3389, ... The earliest negative term (besides a(2)) is a(48673) = -1. Conjecturally infinitely many terms should be negative.

%C Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

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

%F a(n) = -Sum_{primes p<=n} Legendre(p,7) = -Sum_{primes p<=n} Kronecker(-7,p) = -Sum_{primes p<=n} A175629(p).

%e Below 100, there are 10 primes congruent to 1, 2, 4 modulo 7 and 14 primes congruent to 3, 5, 6 modulo 7, so a(100) = 14 - 10 = 4.

%t Accumulate[Table[Which[PrimeQ[n]&&MemberQ[{3,5,6},Mod[n,7]],1,PrimeQ[ n] && MemberQ[ {1,2,4},Mod[ n,7]],-1,True,0],{n,90}]] (* _Harvey P. Dale_, Apr 28 2022 *)

%o (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-7, i))

%Y Cf. A175629.

%Y Let d be a fundamental discriminant.

%Y Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), this sequence (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).

%Y Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

%K sign

%O 1,17

%A _Jianing Song_, Nov 20 2018