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A321856 Number of primes of the form 3*m + 2 <= n minus number of primes of the form 3*m + 1 <= n. 14
0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n) is the number of primes <= n that are quadratic nonresidues modulo 3 minus the number of primes <= n that are quadratic residues modulo 3.

Conjecturally infinitely many terms are negative. The earliest negative term is a(608981813029) = -1, see A112632.

In general, assuming the strong form of RH, if 0 < a, b < k are integers, 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".

LINKS

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

Wikipedia, Chebyshev's bias

FORMULA

a(n) = -Sum_{primes p<=n} Legendre(p,3) = -Sum_{primes p<=n} Kronecker(-3,p) = -Sum_{primes p<=n} A102283(p).

EXAMPLE

Below 100, there are 11 primes congruent to 1 modulo 3 and 13 primes congruent to 2 modulo 3, so a(100) = 13 - 11 = 2.

PROG

(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-3, i))

CROSSREFS

Cf. A102283.

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), this sequence (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: A128016 A128017 A096285 * A175078 A121561 A078772

Adjacent sequences:  A321853 A321854 A321855 * A321857 A321858 A321859

KEYWORD

sign

AUTHOR

Jianing Song, Nov 20 2018

STATUS

approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)