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A057794
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(Integer nearest R(10^n)) - pi(10^n), where pi(x) is the number of primes <= x, R(x) = Sum_{ k>=1 } (mu(k)/k * li(x^(1/k))) and li(x) is the Cauchy principal value of the integral from 0 to x of dt/log(t).
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7
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1, 1, 0, -2, -5, 29, 88, 97, -79, -1828, -2318, -1476, -5773, -19200, 73218, 327052, -598255, -3501366, 23884333, -4891825, -86432204, -127132665, 1033299853, -1658989719
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OFFSET
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1,4
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COMMENTS
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This is Riemann's remarkable approximation for the number of primes <= x.
Equivalently, R(x) is given by the Gram series, 1 + sum of log(x)^k/(k*k!*zeta(k+1)) for k = 1 to infinity. This series converges more quickly.
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REFERENCES
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John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 146.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 90.
Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, http://www.phy.bg.ac.rs/~mphys6/proceedings5/14-KaratsubaKaratsuba.ps.
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LINKS
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Table of n, a(n) for n=1..24.
Tomas Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Eric Weisstein's World of Mathematics, Prime Counting Function.
Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.
Eric Weisstein's World of Mathematics, Gram Series.
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MATHEMATICA
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R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Abs[Round[R[10^n]-PrimePi[10^n]]]
gram[x_] := 1+Sum[N[Log[x]^k/(k*k!*Zeta[k+1]), 100], {k, 1, 1000}]; a[n_] := Abs[Round[gram[10^n]-PrimePi[10^n]]]
(* From version 7 on : *) a[n_] := Round[RiemannR[10^n]-PrimePi[10^n]] (* Jean-François Alcover, Sep 17 2012 *)
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PROG
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(PARI) A057794=vector(#A006880, i, round(1+suminf(k=1, log(10^i)^k/(k*k!*zeta(k+1)))-A006880[i])) \\ - M. F. Hasler, Feb 26 2008
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CROSSREFS
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Cf. A006880, A057752.
Sequence in context: A193901 A083472 A213996 * A073715 A104083 A007014
Adjacent sequences: A057791 A057792 A057793 * A057795 A057796 A057797
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KEYWORD
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sign
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AUTHOR
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Robert G. Wilson v, Nov 04 2000
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EXTENSIONS
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First term corrected by David Baugh, Nov 15 2002
Signs added by M. F. Hasler, Feb 26 2008
The value of a(23) is not known at present, I believe. - N. J. A. Sloane, Mar 17 2008
Last two terms a(23) and a(24), with Pi(10^n) for n=23 and 24 from A006880, from Vladimir Pletser, Feb 27 2013
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STATUS
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approved
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