%I #26 Jul 19 2024 08:46:07

%S 0,-1,0,1,0,1,0,-1,0,1,2,3,2,1,2,3,2,3,2,3,2,3,2,1,0,1,2,1,2,1,2,1,0,

%T -1,0,1,2,1,2,3,2,3,4,3,4,5,4,5,4,5,4,5,4,3,2,3,4,5,6,5,4,5,4,5,4,5,4,

%U 3,2,3,2,3,4,5,4,5,6,7,6,5,4,5,6,5,6,5,4

%N a(n) = A320857(prime(n)).

%C Among the first 10000 terms there are only 100 negative ones. See the comments about "Chebyshev's bias" in A320857.

%H Amiram Eldar, <a href="/A320858/b320858.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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)).

%e prime(46) = 199, Pi(8,1)(199) = 8, Pi(8,5)(199) = 13, Pi(8,3)(199) = Pi(8,7)(199) = 12, so a(46) = 13 + 12 - 8 - 12 = 5.

%t a[n_] := -Sum[KroneckerSymbol[-2, Prime[i]], {i, 1, n}];

%t Array[a, 100] (* _Jean-François Alcover_, Dec 28 2018, from PARI *)

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

%Y Cf. A188510.

%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), this sequence (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

%K sign

%O 1,11

%A _Jianing Song_, Nov 24 2018