

A089896


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


1



6, 30, 177, 1246, 9629, 78627, 664918, 5762209, 50849234, 455055614, 4118066400, 37607950280, 346065645810, 3204942065691, 29844571475287, 279238344248556, 2623557165610821, 24739954309690415, 234057667376222382
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OFFSET

1,1


COMMENTS

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!.
See A057754 for the round() variant. [From R. J. Mathar, Oct 09 2010]


LINKS

Table of n, a(n) for n=1..19.
Xavier Gourdon, Collection of approximations for pi


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 fast converging series taken from the link for the EulerMascheroni 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.


MATHEMATICA

Table[Floor[LogIntegral[10^n]], {n, 19}] (* Arkadiusz Wesolowski, Dec 23 2011 *)


PROG

(PARI) pw2pix(n, m) = { for(x=1, n, y=10^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) }


CROSSREFS

Sequence in context: A175925 A110706 A001341 * A057754 A001473 A334288
Adjacent sequences: A089893 A089894 A089895 * A089897 A089898 A089899


KEYWORD

nonn


AUTHOR

Cino Hilliard, Jan 10 2004


STATUS

approved



