A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.

0, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 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 11 minus the number of primes <= n that are quadratic residues modulo 11.

It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 138 ones are 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,11) = -Sum_{primes p<=n} Kronecker(-11,p) = -Sum_{primes p<=n} A011582(p).

Example

Below 200, there are 20 primes congruent to 1, 3, 4, 5, 9 modulo 11 and 23 primes congruent to 2, 6, 7, 8, 10 modulo 11, so a(200) = 23 - 20 = 3.

Prog

(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-11, i))

Crossrefs

Cf. A112632.

Let d be a fundamental discriminant.

Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: this sequence (d=-11), A320857 (d=-8), A321859 (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: A334138 A210480 A321859 * A348459 A266123 A115230

Adjacent sequences: A321857 A321858 A321859 * A321861 A321862 A321863

Keyword

sign

Author

Jianing Song, Nov 20 2018

Status

approved