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

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

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.

Mathematica

A321860[n_] := -Sum[JacobiSymbol[Prime[i], 11], {i, PrimePi[n]}]; Array[A321860, 100] (* or *)
-Accumulate[Table[If[PrimeQ[n], JacobiSymbol[n, 11], 0], {n, 100}]] (* Paolo Xausa, Jan 09 2026 *)

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