OFFSET
1,5
COMMENTS
a(n) is the number of odd primes <= n that have 2 as a quadratic nonresidue minus the number of primes <= n that have 2 as a quadratic residue. See the comments about "Chebyshev's bias" in A321861. - Jianing Song, Nov 24 2018
Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?
The first negative term occurs at a(11100143) = -1. - Jianing Song, Nov 08 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Wikipedia, Chebyshev's bias
FORMULA
a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018
MATHEMATICA
Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *)
PROG
(PARI) for(n=1, 200, print1(sum(i=1, n, if((i*isprime(i)-3)%8, 0, 1)+if((i*isprime(i)-5)%8, 0, 1)-if((i*isprime(i)-1)%8, 0, 1)-if((i*isprime(i)-7)%8, 0, 1)), ", ")) \\ Program fixed by Jianing Song, Nov 08 2019
(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018
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), this sequence (d=8), A321858 (d=12).
KEYWORD
easy,sign
AUTHOR
Benoit Cloitre, Jun 08 2002
EXTENSIONS
Edited by Peter Munn, Nov 19 2023
STATUS
approved