%I #20 Nov 19 2023 21:16:31

%S 0,1,2,1,2,3,2,3,2,3,2,3,2,3,2,3,4,5,6,5,4,3,4,3,2,3,2,3,4,3,2,3,2,3,

%T 4,3,4,5,4,5,6,7,6,5,6,5,6,5,6,7,6,5,4,5,4,3,4,3,4,3,4,5,6,5,4,5,6,5,

%U 6,7,6,5,4,5,6,5,6,7,6,5,6,7,6,5,4,5,4

%N a(n) = A071838(prime(n)).

%C a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.

%C The first negative term occurs at a(732722) = -1. - _Jianing Song_, Nov 08 2019

%C In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by _Peter Munn_, Nov 19 2023].

%C Here, although 7 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n).

%H Andrew Granville and Greg Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

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

%F a(n) = -Sum_{i=1..n} Kronecker(prime(i),2) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A091337(prime(i)).

%e prime(25) = 97, Pi(8,1)(97) = 5, Pi(8,3)(97) = 7, Pi(8,5)(97) = Pi(8,7)(97) = 6, so a(25) = 7 + 6 - 5 - 6 = 2.

%o (PARI) a(n) = -sum(i=1, n, kronecker(2, prime(i)))

%Y Cf. A007350, A091337.

%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), A321859 (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), this sequence (d=8), A321863 (d=12).

%K sign

%O 1,3

%A _Jianing Song_, Nov 20 2018

%E Edited by _Peter Munn_, Nov 19 2023