# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a071838 Showing 1-1 of 1 %I A071838 #34 Nov 19 2023 21:16:48 %S A071838 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 A071838 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 A071838 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 A071838 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 A071838 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 A071838 Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1? %C A071838 The first negative term occurs at a(11100143) = -1. - _Jianing Song_, Nov 08 2019 %H A071838 Vincenzo Librandi, Table of n, a(n) for n = 1..10000 %H A071838 Wikipedia, Chebyshev's bias %F A071838 a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - _Jianing Song_, Nov 20 2018 %t A071838 Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* _Michael De Vlieger_, Nov 25 2018 *) %o A071838 (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 A071838 (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ _Jianing Song_, Nov 24 2018 %Y A071838 Cf. A091337. %Y A071838 Let d be a fundamental discriminant. %Y A071838 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 A071838 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 A071838 easy,sign %O A071838 1,5 %A A071838 _Benoit Cloitre_, Jun 08 2002 %E A071838 Edited by _Peter Munn_, Nov 19 2023 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE