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 A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. 14
 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) is the number of primes <= n that are quadratic nonresidues modulo 5 minus the number of primes <= n that are quadratic residues modulo 5. a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative. What's the first n such that a(n) < 0? In general, assuming the strong form of RH, 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 n, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". The first negative term occurs at a(2082927221) = -1. - Jianing Song, Nov 08 2019 LINKS Wikipedia, Chebyshev's bias FORMULA a(n) = -Sum_{primes p<=n} Legendre(p,5) = -Sum_{primes p<=n} Kronecker(5,p) = -Sum_{primes p<=n} A080891(p). EXAMPLE Pi(5,1)(100) = Pi(5,4)(100) = 5, Pi(5,2)(100) = Pi(5,3)(100) = 7, so a(100) = 7 + 7 - 5 - 5 = 4. PROG (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(5, i)) CROSSREFS Cf. A080891. 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), this sequence (d=5), A071838 (d=8), A321858 (d=12). 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). Sequence in context: A092363 A133874 A053384 * A186313 A165020 A235224 Adjacent sequences:  A321854 A321855 A321856 * A321858 A321859 A321860 KEYWORD sign AUTHOR Jianing Song, Nov 20 2018 STATUS approved

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Last modified July 10 02:05 EDT 2020. Contains 335570 sequences. (Running on oeis4.)