A320858 a(n) = A320857(prime(n)).

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

Offset

1,11

Comments

Among the first 10000 terms there are only 100 negative ones. See the comments about "Chebyshev's bias" in A320857.

Links

Table of n, a(n) for n=1..87.

Wikipedia, Chebyshev's bias

Formula

a(n) = -Sum_{i=1..n} Kronecker(prime(i),2) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A091337(prime(i)).

Example

prime(46) = 199, Pi(8,1)(199) = 8, Pi(8,5)(199) = 13, Pi(8,3)(199) = Pi(8,7)(199) = 12, so a(46) = 13 + 12 - 8 - 12 = 5.

Mathematica

a[n_] := -Sum[KroneckerSymbol[-2, Prime[i]], {i, 1, n}];
Array[a, 100] (* Jean-François Alcover, Dec 28 2018, from PARI *)

Prog

(PARI) a(n) = -sum(i=1, n, kronecker(-2, prime(i)))

Crossrefs

Cf. A188510.

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), this sequence (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Sequence in context: A084199 A277745 A353497 * A304111 A030314 A269624

Adjacent sequences: A320855 A320856 A320857 * A320859 A320860 A320861

Keyword

sign

Author

Jianing Song, Nov 24 2018

Status

approved