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 A057794 (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). 11
 1, 1, 0, -2, -5, 29, 88, 97, -79, -1828, -2318, -1476, -5773, -19200, 73218, 327052, -598255, -3501366, 23884333, -4891825, -86432204, -127132665, 1033299853, -1658989719 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. REFERENCES 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. LINKS Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 14-23 September 2010, Belgrade, Serbia. [From Internet Archive Wayback Machine] Tomás 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. MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A006880, A057752. Sequence in context: A193901 A083472 A213996 * A257545 A073715 A104083 Adjacent sequences:  A057791 A057792 A057793 * A057795 A057796 A057797 KEYWORD sign AUTHOR Robert G. Wilson v, Nov 04 2000 EXTENSIONS 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 STATUS approved

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Last modified July 22 14:46 EDT 2019. Contains 325224 sequences. (Running on oeis4.)