

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.


14



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, 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, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4
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OFFSET

1,3


COMMENTS

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.
a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative. What's the first n such that a(n) < 0?
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".
The first negative term occurs at a(2082927221) = 1.  Jianing Song, Nov 08 2019


LINKS

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


FORMULA

a(n) = Sum_{primes p<=n} Legendre(p,5) = Sum_{primes p<=n} Kronecker(5,p) = Sum_{primes p<=n} A080891(p).


EXAMPLE

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.


PROG

(PARI) a(n) = sum(i=1, n, isprime(i)*kronecker(5, i))


CROSSREFS

Cf. A080891.
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), this sequence (d=5), A071838 (d=8), A321858 (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: A092363 A133874 A053384 * A186313 A165020 A235224
Adjacent sequences: A321854 A321855 A321856 * A321858 A321859 A321860


KEYWORD

sign


AUTHOR

Jianing Song, Nov 20 2018


STATUS

approved



