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A321856
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Number of primes of the form 3*m + 2 <= n minus number of primes of the form 3*m + 1 <= n.
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17
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0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2
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OFFSET
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1,5
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COMMENTS
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a(n) is the number of primes <= n that are quadratic nonresidues modulo 3 minus the number of primes <= n that are quadratic residues modulo 3.
Conjecturally infinitely many terms are negative. The earliest negative term is a(608981813029) = -1, see A112632.
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. 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". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 05 2023]
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LINKS
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Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
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FORMULA
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a(n) = -Sum_{primes p<=n} Legendre(p,3) = -Sum_{primes p<=n} Kronecker(-3,p) = -Sum_{primes p<=n} A102283(p).
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EXAMPLE
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Below 100, there are 11 primes congruent to 1 modulo 3 and 13 primes congruent to 2 modulo 3, so a(100) = 13 - 11 = 2.
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PROG
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(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-3, 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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STATUS
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approved
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