A321859 Number of primes congruent to 3, 5, 6 modulo 7 and <= n minus number of primes congruent to 1, 2, 4 modulo 7 and <= n.

0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2

Offset

1,17

Comments

a(n) is the number of primes <= n that are quadratic nonresidues modulo 7 minus the number of primes <= n that are quadratic residues modulo 7.

The first 10000 terms (except for a(2)) are nonnegative. a(p) = 0 for primes p = 3, 11, 211, 3371, 3389, ... The earliest negative term (besides a(2)) is a(48673) = -1. Conjecturally infinitely many terms should be negative.

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

Links

Table of n, a(n) for n=1..87.

Wikipedia, Chebyshev's bias

Formula

a(n) = -Sum_{primes p<=n} Legendre(p,7) = -Sum_{primes p<=n} Kronecker(-7,p) = -Sum_{primes p<=n} A175629(p).

Example

Below 100, there are 10 primes congruent to 1, 2, 4 modulo 7 and 14 primes congruent to 3, 5, 6 modulo 7, so a(100) = 14 - 10 = 4.

Mathematica

Accumulate[Table[Which[PrimeQ[n]&&MemberQ[{3, 5, 6}, Mod[n, 7]], 1, PrimeQ[ n] && MemberQ[ {1, 2, 4}, Mod[ n, 7]], -1, True, 0], {n, 90}]] (* Harvey P. Dale, Apr 28 2022 *)

Prog

(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-7, i))

Crossrefs

Cf. A175629.

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), this sequence (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (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: A253315 A334138 A210480 * A321860 A348459 A266123

Adjacent sequences: A321856 A321857 A321858 * A321860 A321861 A321862

Keyword

sign

Author

Jianing Song, Nov 20 2018

Status

approved