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A071838 a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. 14
0, 0, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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. - Jianing Song, Nov 24 2018

Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?

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". - Jianing Song, Nov 24 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Wikipedia, Chebyshev's bias

FORMULA

a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018

MATHEMATICA

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *)

PROG

(PARI) for(n=1, 200, print1(sum(i=1, n, if((i*isprime(i)-3)%8, 0, 1)+if((i*isprime(i)-5)%8, 0, 1))-if((i*isprime(i)-1)%8, 0, 1))-if((i*isprime(i)-7)%8, 0, 1)), ", "))

(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018

CROSSREFS

Cf. A091337.

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), this sequence (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: A238643 A140193 A073741 * A157896 A156072 A215788

Adjacent sequences:  A071835 A071836 A071837 * A071839 A071840 A071841

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Jun 08 2002

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)