A321864 a(n) = A321859(prime(n)).

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

Offset

1,7

Comments

Among the first 10000 terms there are only 13 negative ones, with the earliest one (besides a(1)) being a(5006) = -1.

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(prime(i),7) = -Sum_{primes p<=n} Kronecker(-7,prime(i)) = -Sum_{i=1..n} A175629(prime(i)).

Example

prime(25) = 97. Among the primes <= 97, there are 10 ones congruent to 1, 2, 4 modulo 7 and 14 ones congruent to 3, 5, 6 modulo 7, so a(25) = 14 - 10 = 4.

Prog

(PARI) a(n) = -sum(i=1, n, kronecker(-7, prime(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), 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), this sequence (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Sequence in context: A277914 A126626 A137927 * A084311 A353486 A026490

Adjacent sequences: A321861 A321862 A321863 * A321865 A321866 A321867

Keyword

sign

Author

Jianing Song, Nov 20 2018

Status

approved