A321862 - OEIS

%I #18 Nov 19 2023 10:23:46

%S 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,

%T 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,

%U 8,7,8,7,8,9,8,9,8,9,8,7,6,5,4,5,4,5,4

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

%C The first 10000 terms are positive, but conjecturally infinitely many terms should be negative.

%C The first negative term occurs at a(102091236) = -1. - _Jianing Song_, Nov 08 2019

%C Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

%F 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)).

%e 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.

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

%Y Cf. A080891.

%Y Let d be a fundamental discriminant.

%Y 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).

%Y 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).

%K sign

%O 1,2

%A _Jianing Song_, Nov 20 2018

%E Edited by _Peter Munn_, Nov 19 2023