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A089897 a(n) = floor(Li(2^n)), where Li(x) is the integral from 0 to x of dt/log(t).

%I A089897 #19 Mar 08 2025 02:28:04
%S A089897 1,2,5,8,13,21,36,60,103,181,321,576,1047,1919,3544,6583,12296,23069,
%T A089897 43453,82137,155739,296113,564411,1078221,2063984,3958349,7604383,
%U A089897 14631777,28194305,54401475,105100230,203284081,393619392,762944445
%N A089897 a(n) = floor(Li(2^n)), where Li(x) is the integral from 0 to x of dt/log(t).
%C A089897 a(n) approximates A007053(n).
%H A089897 Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/constants.html">Collection of approximations for pi</a>
%F A089897 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 rapidly 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 A089897 Table[ Floor[ LogIntegral[2^n]], {n, 34}] (* _Robert G. Wilson v_, Nov 09 2005 *)
%o A089897 (PARI)
%o A089897 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 A089897 pw2pix(n,m) = { for(x=1,n, y=2^x; print1(floor(Li(y,m))",") ) }
%K A089897 nonn
%O A089897 1,2
%A A089897 _Cino Hilliard_, Jan 10 2004