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.

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

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.

The first negative term occurs at a(2082927221) = -1. - Jianing Song, Nov 08 2019

Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

Assuming the generalized Riemann hypothesis (GRH) and the Grand Simplicity Hypothesis (GSH; called LI in page 29 of Greg Martin's 2010 link), the set {x>1: pi_{5,2}(x) + pi_{5,3}(x) > pi_{5,1}(x) + pi_{5,4}(x)} has logarithmic density of about 0.9521 (see page 117 of Greg Martin's 2018 link below; the assertion is written as mod 10 which makes no difference). - Jianing Song, Jan 08 2026

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Greg Martin and Daniel Fiorilli, Prime number races: An asymptotic formula for the densities, Analytic and Combinatorial Number Theory, Institute of Mathematical Sciences, Chennai, India. August 31, 2010.

Greg Martin, Prime number races, Pure Math Seminar, UNSW Canberra. November 29, 2018.

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.

Mathematica

A321857[n_] := -Sum[JacobiSymbol[Prime[i], 5], {i, PrimePi[n]}]; Array[A321857, 100] (* or *)
-Accumulate[Table[If[PrimeQ[n], JacobiSymbol[n, 5], 0], {n, 100}]] (* Paolo Xausa, Jan 09 2026 *)

Prog

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

Crossrefs

Prime indices of -1, 0, 1, 2, 3: A392295, A392296, A392297, A392298, A392299.

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

Extensions

Edited by Peter Munn, Nov 18 2023

Status

approved