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%I #15 Jul 19 2015 01:29:48
%S 1,2,5,8,13,21,36,60,103,181,321,576,1047,1919,3544,6583,12296,23069,
%T 43453,82137,155739,296113,564411,1078221,2063984,3958349,7604383,
%U 14631777,28194305,54401475,105100230,203284081,393619392,762944445
%N a(n) = floor(Li(2^n)), where Li(x) is the integral from 0 to x of dt/log(t).
%C a(n) approximates A007053(n).
%H Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/constants.html">Collection of approximations for pi</a>
%F 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 Table[ Floor[ LogIntegral[2^n]], {n, 34}] (* _Robert G. Wilson v_, Nov 09 2005 *)
%o (PARI) pw2pix(n,m) = { for(x=1,n, y=2^x; print1(floor(Li(y,m))",") ) } 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) }
%K nonn
%O 1,2
%A _Cino Hilliard_, Jan 10 2004
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