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A320857 a(n) = Pi(8,5)(n) + Pi(8,7)(n) - Pi(8,1)(n) - Pi(8,3)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. 13
0, 0, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,31

COMMENTS

a(n) is the number of odd primes <= n that have -2 as a quadratic nonresidue minus the number of primes <= n that have -2 as a quadratic residue.

It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 212 ones are negative.

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. 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_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).

EXAMPLE

Pi(8,1)(200) = 8, Pi(8,5)(200) = 13, Pi(8,3)(200) = Pi(8,7)(200) = 12, so a(200) = 13 + 12 - 8 - 12 = 5.

MATHEMATICA

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)

PROG

(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-2, 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), this sequence (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), A321863 (d=12).

Sequence in context: A095139 A109038 A208246 * A211664 A182434 A185679

Adjacent sequences:  A320854 A320855 A320856 * A320858 A320859 A320860

KEYWORD

sign

AUTHOR

Jianing Song, Nov 24 2018

STATUS

approved

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Last modified December 8 17:36 EST 2019. Contains 329865 sequences. (Running on oeis4.)