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A321859 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.

%I A321859 #17 Nov 19 2023 10:23:55
%S A321859 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 A321859 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 A321859 2,2,1,1,2,2,2,2,2,2,1,1,1,1,2,2,2,2,2
%N A321859 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 A321859 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 A321859 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 A321859 Please see the comment in A321856 describing "Chebyshev's bias" in the general case.
%H A321859 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%F A321859 a(n) = -Sum_{primes p<=n} Legendre(p,7) = -Sum_{primes p<=n} Kronecker(-7,p) = -Sum_{primes p<=n} A175629(p).
%e A321859 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 A321859 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 A321859 (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-7, i))
%Y A321859 Cf. A175629.
%Y A321859 Let d be a fundamental discriminant.
%Y A321859 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 A321859 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 A321859 sign
%O A321859 1,17
%A A321859 _Jianing Song_, Nov 20 2018