

A321861


a(n) = A071838(prime(n)).


13



0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 5, 6, 5, 4, 3, 4, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 7, 6, 5, 4, 5, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 6, 7, 6, 5, 4, 5, 4
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OFFSET

1,3


COMMENTS

a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative. What's the first n such that a(n) < 0?
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". Here, although 7 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(25) = 97, Pi(8,1)(97) = 5, Pi(8,3)(97) = 7, Pi(8,5)(97) = Pi(8,7)(97) = 6, so a(25) = 7 + 6  5  6 = 2.


PROG

(PARI) a(n) = sum(i=1, n, kronecker(2, prime(i)))


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), 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), this sequence (d=8), A321863 (d=12).
Sequence in context: A105499 A005812 A136625 * A283431 A258594 A086520
Adjacent sequences: A321858 A321859 A321860 * A321862 A321863 A321864


KEYWORD

nonn


AUTHOR

Jianing Song, Nov 20 2018


STATUS

approved



