login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A320858 a(n) = A320857(prime(n)). 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

Among the first 10000 terms there are only 100 negative ones.

In general, assuming the strong form of RH, if 0 < a, b < k, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod n, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". Here, although 3 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n).

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-Fran├žois 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: A032452 A084199 A277745 * A304111 A030314 A269624

Adjacent sequences:  A320855 A320856 A320857 * A320859 A320860 A320861

KEYWORD

sign

AUTHOR

Jianing Song, Nov 24 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 16 21:59 EST 2019. Contains 320200 sequences. (Running on oeis4.)