# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a321858 Showing 1-1 of 1 %I A321858 #21 Nov 19 2023 10:23:31 %S A321858 0,0,0,0,1,1,2,2,2,2,1,1,0,0,0,0,1,1,2,2,2,2,1,1,1,1,1,1,2,2,3,3,3,3, %T A321858 3,3,2,2,2,2,3,3,4,4,4,4,3,3,3,3,3,3,4,4,4,4,4,4,3,3,2,2,2,2,2,2,3,3, %U A321858 3,3,2,2,1,1,1,1,1,1,2,2,2,2,1,1,1,1,1 %N A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(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 A321858 a(n) is the number of odd primes <= n that have 3 as a quadratic nonresidue minus the number of primes <= n that have 3 as a quadratic residue. %C A321858 The first 10000 terms are nonnegative. a(p) = 0 for primes p = 2, 3, 13, 433, 443, 457, 479, 491, 503, 3541, ... The earliest negative term is a(61463) = -1. Conjecturally infinitely many terms should be negative. %C A321858 In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by _Peter Munn_, Nov 19 2023] %C A321858 Here, although 11 is not a quadratic residue modulo 12, for most n we have Pi(12,7)(n) + Pi(12,11)(n) > Pi(12,1)(n) - Pi(12,5)(n), Pi(12,5)(n) + Pi(12,11)(n) > Pi(12,1)(n) + Pi(12,7)(n) and Pi(12,5)(n) + Pi(12,7)(n) > Pi(12,1)(n) + Pi(12,11)(n). %H A321858 Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33. %H A321858 Wikipedia, Chebyshev's bias %F A321858 a(n) = -Sum_{primes p<=n} Kronecker(12,p) = -Sum_{primes p<=n} A110161(p). %e A321858 Pi(12,1)(100) = 5, Pi(12,5)(100) = Pi(12,7)(100) = Pi(12,11)(100) = 6, so a(100) = 6 + 6 - 5 - 6 = 1. %o A321858 (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(12, i)) %Y A321858 Cf. A007350, A110161. %Y A321858 Let d be a fundamental discriminant. %Y A321858 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), this sequence (d=12). %Y A321858 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 A321858 sign %O A321858 1,7 %A A321858 _Jianing Song_, Nov 20 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE