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 A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n. 15
 0, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,17 COMMENTS a(n) is the number of primes <= n that are quadratic nonresidues modulo 11 minus the number of primes <= n that are quadratic residues modulo 11. It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 138 ones are negative. 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". LINKS Wikipedia, Chebyshev's bias FORMULA a(n) = -Sum_{primes p<=n} Legendre(p,11) = -Sum_{primes p<=n} Kronecker(-11,p) = -Sum_{primes p<=n} A011582(p). EXAMPLE Below 200, there are 20 primes congruent to 1, 3, 4, 5, 9 modulo 11 and 23 primes congruent to 2, 6, 7, 8, 10 modulo 11, so a(200) = 23 - 20 = 3. PROG (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-11, i)) CROSSREFS Cf. A112632. Let d be a fundamental discriminant. Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: this sequence (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), A321861 (d=8), A321863 (d=12). Sequence in context: A334138 A210480 A321859 * A266123 A115230 A338686 Adjacent sequences:  A321857 A321858 A321859 * A321861 A321862 A321863 KEYWORD sign AUTHOR Jianing Song, Nov 20 2018 STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)