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 A321861 a(n) = A071838(prime(n)). 13
 0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 5, 6, 5, 4, 3, 4, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 7, 6, 5, 4, 5, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 6, 7, 6, 5, 4, 5, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". Here, although 7 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n). LINKS Wikipedia, Chebyshev's bias FORMULA a(n) = -Sum_{i=1..n} Kronecker(prime(i),2) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A091337(prime(i)). EXAMPLE prime(25) = 97, Pi(8,1)(97) = 5, Pi(8,3)(97) = 7, Pi(8,5)(97) = Pi(8,7)(97) = 6, so a(25) = 7 + 6 - 5 - 6 = 2. PROG (PARI) a(n) = -sum(i=1, n, kronecker(2, prime(i))) 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), 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), this sequence (d=8), A321863 (d=12). Sequence in context: A105499 A005812 A136625 * A283431 A258594 A086520 Adjacent sequences:  A321858 A321859 A321860 * A321862 A321863 A321864 KEYWORD nonn AUTHOR Jianing Song, Nov 20 2018 STATUS approved

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Last modified October 16 13:32 EDT 2019. Contains 328093 sequences. (Running on oeis4.)