

A321862


a(n) = A321857(prime(n)).


13



1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 3, 4, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 5, 4
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OFFSET

1,2


COMMENTS

The first 10000 terms are positive, 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".


LINKS

Table of n, a(n) for n=1..87.
Wikipedia, Chebyshev's bias


FORMULA

a(n) = Sum_{i=1..n} Legendre(prime(i),5) = Sum_{primes p<=n} Kronecker(2,prime(i)) = Sum_{i=1..n} A080891(prime(i)).


EXAMPLE

prime(25) = 97, Pi(5,1)(97) = Pi(5,4)(97) = 5, Pi(5,2)(97) = Pi(5,3)(97) = 7, so a(25) = 7 + 7  5  5 = 4.


PROG

(PARI) a(n) = sum(i=1, n, kronecker(5, prime(i)))


CROSSREFS

Cf. A080891.
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), this sequence (d=5), A321861 (d=8), A321863 (d=12).
Sequence in context: A131830 A147952 A091316 * A071825 A115727 A115726
Adjacent sequences: A321859 A321860 A321861 * A321863 A321864 A321865


KEYWORD

nonn


AUTHOR

Jianing Song, Nov 20 2018


STATUS

approved



