

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
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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



