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%I #19 Sep 08 2022 08:45:02
%S 6,30,178,1246,9630,78628,664918,5762209,50849235,455055615,
%T 4118066401,37607950281,346065645810,3204942065692,29844571475288,
%U 279238344248557,2623557165610822,24739954309690415,234057667376222382
%N Integer nearest to Li(10^n), where Li(x) = integral(0..x, dt/log(t)).
%C "Li[z] is central to the study of the distribution of primes in number theory. The logarithmic integral function is sometimes also denoted by Li(z). In some number-theoretical applications li(z) is defined as [integral from 2 to z of 1/log(t) dt], with no principal value taken. This differs from the definition used in 'Mathematica' by the constant li(2)."
%H Vincenzo Librandi, <a href="/A057754/b057754.txt">Table of n, a(n) for n = 1..1000</a>
%H C. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml#table">values of pi(x)</a>
%H B. Riemann, <a href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/">On the Number of Prime Numbers</a> 1859, last page (various transcripts)
%H Stephen Wolfram, <a href="http://documents.wolfram.com/v3/">The Mathematica 3 Book</a>, 1996, Section 3.2.10: Special Functions.
%F a(n) = round( Li( 10^n )) = round( Ei( log( 10^n ))).
%e Li( 10^22 ) = 201467286691248261498.15... => a(22).
%e pi( 10^22 ) = 201467286689315906290.
%p seq(round(evalf(Li(10^n), 64)), n=1..19); # _Peter Luschny_, Mar 20 2019
%t Table[Round[LogIntegral[10^n]], {n, 1, 25}]
%o (PARI) vector(25, n, round(real(-eint1(-log(10^n)))) ) \\ _G. C. Greubel_, May 17 2019
%o (Magma) [Round(LogIntegral(10^n)): n in [1..25]]; // _G. C. Greubel_, May 17 2019
%o (Sage) [round(li(10^n)) for n in (1..25)] # _G. C. Greubel_, May 17 2019
%Y A052435( 10^n ) = a(n) - pi( 10^n ) for n > 0.
%Y Cf. A000720, A007504.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Oct 30 2000
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