%I A057794
%S A057794 1,1,0,2,5,29,88,97,79,1828,2318,1476,5773,19200,73218,
%T A057794 327052,598255,3501366,23884333,4891825,86432204,127132665
%V A057794 1,1,0,-2,-5,29,88,97,-79,-1828,-2318,-1476,-5773,-19200,73218,
%W A057794 327052,-598255,-3501366,23884333,-4891825,-86432204,-127132665
%N 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).
%C A057794 This is Riemann's remarkable approximation for the number of primes <= 
               x.
%C A057794 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.
%D A057794 John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint 
               of Springer-Verlag, NY, 1996, page 146.
%D A057794 M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 
               2003; see p. 90.
%H A057794 Tomas Oliveira e Silva, <a href="http://www.ieeta.pt/~tos/primes.html">
               Tables of values of pi(x) and of pi2(x)</a>
%H A057794 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeCountingFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A057794 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannPrimeNumberFormula.html">Link to a section of The World of 
               Mathematics.</a>
%H A057794 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GramSeries.html">Link to a section of The World of Mathematics.</
               a>
%t A057794 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]]]
%t A057794 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]]]
%o A057794 (PARI) A057794=vector(#A006880,i,round(1+suminf(k=1, log(10^i)^k/(k*k!*zeta(k+1)))-A006880[i])) 
               \\ - M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 26 2008
%Y A057794 Cf. A006880, A057752.
%Y A057794 Sequence in context: A061351 A126107 A083472 this_sequence A073715 A104083 
               A132282
%Y A057794 Adjacent sequences: A057791 A057792 A057793 this_sequence A057795 A057796 
               A057797
%K A057794 sign
%O A057794 1,4
%A A057794 Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2000
%E A057794 First term corrected by David Baugh (dbaugh(AT)owlnet.rice.edu), Nov 
               15 2002
%E A057794 Signs added by M. F. Hasler, Feb 26 2008
%E A057794 The value of a(23) is not known at present, I believe. - N. J. A. Sloane 
               (njas(AT)research.att.com), Mar 17 2008