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A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

%I A321857 #19 Nov 18 2023 18:09:13
%S A321857 0,1,2,2,2,2,3,3,3,3,2,2,3,3,3,3,4,4,3,3,3,3,4,4,4,4,4,4,3,3,2,2,2,2,
%T A321857 2,2,3,3,3,3,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,4,4,3,3,3,3,3,3,4,4,
%U A321857 4,4,3,3,4,4,4,4,4,4,3,3,3,3,4,4,4,4,4
%N A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.
%C A321857 a(n) is the number of primes <= n that are quadratic nonresidues modulo 5 minus the number of primes <= n that are quadratic residues modulo 5.
%C A321857 a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.
%C A321857 The first negative term occurs at a(2082927221) = -1. - _Jianing Song_, Nov 08 2019
%C A321857 Please see the comment in A321856 describing "Chebyshev's bias" in the general case.
%H A321857 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%F A321857 a(n) = -Sum_{primes p<=n} Legendre(p,5) = -Sum_{primes p<=n} Kronecker(5,p) = -Sum_{primes p<=n} A080891(p).
%e A321857 Pi(5,1)(100) = Pi(5,4)(100) = 5, Pi(5,2)(100) = Pi(5,3)(100) = 7, so a(100) = 7 + 7 - 5 - 5 = 4.
%o A321857 (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(5, i))
%Y A321857 Cf. A080891.
%Y A321857 Let d be a fundamental discriminant.
%Y A321857 Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), this sequence (d=5), A071838 (d=8), A321858 (d=12).
%Y A321857 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 A321857 sign
%O A321857 1,3
%A A321857 _Jianing Song_, Nov 20 2018
%E A321857 Edited by _Peter Munn_, Nov 18 2023