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A321864 a(n) = A321859(prime(n)). 13

%I

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

%T 2,1,2,1,2,3,2,3,2,1,0,1,0,1,2,3,2,1,2,3,4,3,4,5,4,3,4,5,6,7,8,7,6,5,

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

%N a(n) = A321859(prime(n)).

%C Among the first 10000 terms there are only 13 negative ones, with the earliest one (besides a(1)) being a(5006) = -1.

%C In general, assuming the strong form of RH, 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 n, 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".

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

%F a(n) = -Sum_{primes p<=n} Legendre(prime(i),7) = -Sum_{primes p<=n} Kronecker(-7,prime(i)) = -Sum_{i=1..n} A175629(prime(i)).

%e prime(25) = 97. Among the primes <= 97, there are 10 ones congruent to 1, 2, 4 modulo 7 and 14 ones congruent to 3, 5, 6 modulo 7, so a(25) = 14 - 10 = 4.

%o (PARI) a(n) = -sum(i=1, n, kronecker(-7, prime(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), 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), this sequence (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

%K sign

%O 1,7

%A _Jianing Song_, Nov 20 2018

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)