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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 (list; graph; refs; listen; history; text; internal format)
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,changed

AUTHOR

Jianing Song, Nov 20 2018

STATUS

approved

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Last modified November 14 14:43 EST 2019. Contains 329126 sequences. (Running on oeis4.)