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A321857
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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.
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15
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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
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OFFSET
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1,3
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COMMENTS
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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.
The first negative term occurs at a(2082927221) = -1. - Jianing Song, Nov 08 2019
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.
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LINKS
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FORMULA
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a(n) = -Sum_{primes p<=n} Legendre(p,5) = -Sum_{primes p<=n} Kronecker(5,p) = -Sum_{primes p<=n} A080891(p).
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EXAMPLE
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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.
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PROG
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(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(5, i))
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CROSSREFS
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Let d be a fundamental discriminant.
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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