A321863 a(n) = A321858(prime(n)).

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

Offset: 1

Author

Jianing Song, Nov 20 2018

Keywords

• sign

Comments

Among the first 10000 terms there are only 291 negative ones, with the earliest one being a(6181) = -1. See the comments about "Chebyshev's bias" in A321858.

Examples

			prime(25) = 97, Pi(12,1)(97) = 5, Pi(12,5)(97) = Pi(12,7)(97) = Pi(12,11)(97) = 6, so a(25) = 6 + 6 - 5 - 6 = 1.
		

Links

• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A110161.

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), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), this sequence (d=12).

Programs

PARI

a(n) = -sum(i=1, n, kronecker(12, prime(i)))

Formula

a(n) = -Sum_{primes p<=n} Kronecker(12,prime(i)) = -Sum_{i=1..n} A110161(prime(i)).