A071838 a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(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, 0, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2

Offset: 1

Author

Benoit Cloitre, Jun 08 2002

Keywords

• easy
• sign

Comments

a(n) is the number of odd primes <= n that have 2 as a quadratic nonresidue minus the number of primes <= n that have 2 as a quadratic residue. See the comments about "Chebyshev's bias" in A321861. - Jianing Song, Nov 24 2018

Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?

The first negative term occurs at a(11100143) = -1. - Jianing Song, Nov 08 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A091337.

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), this sequence (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).

Programs

Mathematica

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *)

PARI

for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-3)%8,0,1)+if((i*isprime(i)-5)%8,0,1)-if((i*isprime(i)-1)%8,0,1)-if((i*isprime(i)-7)%8,0,1)),", ")) \\ Program fixed by Jianing Song, Nov 08 2019

PARI

a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018

Formula

a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018

Extensions

Edited by Peter Munn, Nov 19 2023