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A089896 Logarithmic integral approximation to number of primes less than 10^x.

%I A089896 #23 Mar 08 2025 02:36:44
%S A089896 6,30,177,1246,9629,78627,664918,5762209,50849234,455055614,
%T A089896 4118066400,37607950280,346065645810,3204942065691,29844571475287,
%U A089896 279238344248556,2623557165610821,24739954309690415,234057667376222382
%N A089896 Logarithmic integral approximation to number of primes less than 10^x.
%C A089896 In computing Li(x) we can limit the iterations to 2*log(x) + m where m is suitably large to allow convergence to the precision desired. If we let m = floor(log(log(x))) we get a better approximation of Pi(x) than the full Li(x) expansion. With this m we get Li(x) < Pi(x) often but still closer in absolute value to Pi(x). Note the use of the gamma function to quickly compute factorials in the precision range i.e. gamma(x+1) = x!.
%C A089896 See A057754 for the round() variant. [From _R. J. Mathar_, Oct 09 2010]
%H A089896 Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/constants.html">Collection of approximations for pi</a>
%F A089896 The logarithmic Integral can be computed by Li(x) = log(log(x)) + log(x) + log(x)^2/2/2! + log(x)^3/3/3! + ... + 1 - log(3/2) - sum(k=1, prec, (zeta(2k+1)-1)/(2k+1)/4^k). This last expression is a fast converging series taken from the link for the Euler-Mascheroni constant 0.57721.. where prec is the precision level you are using. PARI has an Euler() function built in so that was used in this calculation.
%t A089896 Table[Floor[LogIntegral[10^n]], {n, 19}] (* _Arkadiusz Wesolowski_, Dec 23 2011 *)
%o A089896 (PARI)
%o A089896 Li(n,m) = { y2 = log(n); y = 1; z=1; s=log(y2)+ Euler(); for(x=1,floor(2*log(n)+m),y=y2^x/x/gamma(x+1); s+=y; ); return(s) }
%o A089896 pw2pix(n,m) = { for(x=1,n, y=10^x; print1(floor(Li(y,m))",") ) }
%K A089896 nonn
%O A089896 1,1
%A A089896 _Cino Hilliard_, Jan 10 2004