A321865 a(n) = A321860(prime(n)).

1, 0, -1, 0, 0, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 3, 4, 3, 4, 3, 2

Offset

1,7

Comments

Among the first 10000 terms there are only 32 negative ones.

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(prime(i),11) = -Sum_{primes p<=n} Kronecker(-11,prime(i)) = -Sum_{i=1..n} A011582(prime(i)).

Example

prime(46) = 199. Among the primes <= 199, there are 20 ones congruent to 1, 3, 4, 5, 9 modulo 11 and 23 ones congruent to 2, 6, 7, 8, 10 modulo 11, so a(46) = 23 - 20 = 3.

Mathematica

A321865[n_] := -Sum[JacobiSymbol[Prime[i], 11], {i, n}]; Array[A321865, 100] (* or *)
-Accumulate[JacobiSymbol[Prime[Range[100]], 11]] (* Paolo Xausa, Jan 10 2026 *)

Prog

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

Crossrefs

Cf. A011582.

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), A071838 (d=8), A321858 (d=12).

Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: this sequence (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: A075789 A341018 A214323 * A353526 A244544 A159580

Adjacent sequences: A321862 A321863 A321864 * A321866 A321867 A321868

Keyword

sign

Author

Jianing Song, Nov 20 2018

Status

approved