

A321858


a(n) = Pi(12,5)(n) + Pi(12,7)(n)  Pi(12,1)(n)  Pi(12,11)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.


13



0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1
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OFFSET

1,7


COMMENTS

a(n) is the number of odd primes <= n that have 3 as a quadratic nonresidue minus the number of primes <= n that have 3 as a quadratic residue.
The first 10000 terms are nonnegative. a(p) = 0 for primes p = 2, 3, 13, 433, 443, 457, 479, 491, 503, 3541, ... The earliest negative term is a(61463) = 1. Conjecturally infinitely many terms should be negative.
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. This phenomenon is called "Chebyshev's bias". Here, although 11 is not a quadratic residue modulo 12, for most n we have Pi(12,7)(n) + Pi(12,11)(n) > Pi(12,1)(n)  Pi(12,5)(n), Pi(12,5)(n) + Pi(12,11)(n) > Pi(12,1)(n) + Pi(12,7)(n) and Pi(12,5)(n) + Pi(12,7)(n) > Pi(12,1)(n) + Pi(12,11)(n).


LINKS

Table of n, a(n) for n=1..87.
Wikipedia, Chebyshev's bias


FORMULA

a(n) = Sum_{primes p<=n} Kronecker(12,p) = Sum_{primes p<=n} A110161(p).


EXAMPLE

Pi(12,1)(100) = 5, Pi(12,5)(100) = Pi(12,7)(100) = Pi(12,11)(100) = 6, so a(100) = 6 + 6  5  6 = 1.


PROG

(PARI) a(n) = sum(i=1, n, isprime(i)*kronecker(12, i))


CROSSREFS

Cf. A110161.
Let d be a fundamental discriminant.
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), this sequence (d=12).
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).
Sequence in context: A318722 A174344 A049241 * A230415 A101080 A130836
Adjacent sequences: A321855 A321856 A321857 * A321859 A321860 A321861


KEYWORD

sign


AUTHOR

Jianing Song, Nov 20 2018


STATUS

approved



