|
%I #34 Nov 19 2023 21:16:48
%S 0,0,1,1,2,2,1,1,1,1,2,2,3,3,3,3,2,2,3,3,3,3,2,2,2,2,2,2,3,3,2,2,2,2,
%T 2,2,3,3,3,3,2,2,3,3,3,3,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,6,6,
%U 6,6,5,5,4,4,4,4,4,4,3,3,3,3,4,4,4,4,4,4,3,3,3,3,3,3,3,3,2,2,2,2,3,3,2,2,2
%N a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.
%C 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
%C Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?
%C The first negative term occurs at a(11100143) = -1. - _Jianing Song_, Nov 08 2019
%H Vincenzo Librandi, <a href="/A071838/b071838.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_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - _Jianing Song_, Nov 20 2018
%t Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* _Michael De Vlieger_, Nov 25 2018 *)
%o (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
%o (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ _Jianing Song_, Nov 24 2018
%Y Cf. A091337.
%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), this sequence (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), A321862 (d=5), A321861 (d=8), A321863 (d=12).
%K easy,sign
%O 1,5
%A _Benoit Cloitre_, Jun 08 2002
%E Edited by _Peter Munn_, Nov 19 2023
|
