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A321865 a(n) = A321860(prime(n)). 13

%I

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

%T 3,4,5,4,3,4,5,4,3,2,3,4,3,4,3,4,3,4,5,6,5,4,5,4,5,6,7,8,9,10,9,8,7,6,

%U 7,8,9,8,9,8,9,8,7,6,5,4,5,4,3,4,3,4,3,2

%N a(n) = A321860(prime(n)).

%C Among the first 10000 terms there are only 32 negative ones.

%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),11) = -Sum_{primes p<=n} Kronecker(-11,prime(i)) = -Sum_{i=1..n} A011582(prime(i)).

%e prime(46) = 199. Among the primes <= 199, there are 20 ones congruent to 1, 3, 4, 5, 9 modulo 11 and 23 ones congruent to 2, 6, 7, 8, 10 modulo 11, so a(46) = 23 - 20 = 3.

%o (PARI) a(n) = -sum(i=1, n, kronecker(-11, prime(i)))

%Y Cf. A011582.

%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: this sequence (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 1,7

%A _Jianing Song_, Nov 20 2018

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Last modified April 26 12:07 EDT 2019. Contains 322472 sequences. (Running on oeis4.)