A089897 a(n) = floor(Li(2^n)), where Li(x) is the integral from 0 to x of dt/log(t).
1, 2, 5, 8, 13, 21, 36, 60, 103, 181, 321, 576, 1047, 1919, 3544, 6583, 12296, 23069, 43453, 82137, 155739, 296113, 564411, 1078221, 2063984, 3958349, 7604383, 14631777, 28194305, 54401475, 105100230, 203284081, 393619392, 762944445
Offset: 1
Keywords
Links
- Xavier Gourdon, Collection of approximations for pi
Programs
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Mathematica
Table[ Floor[ LogIntegral[2^n]], {n, 34}] (* Robert G. Wilson v, Nov 09 2005 *) -
PARI
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) } pw2pix(n,m) = { for(x=1,n, y=2^x; print1(floor(Li(y,m))",") ) }
Formula
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.
Comments