A057754 Integer nearest to Li(10^n), where Li(x) = integral(0..x, dt/log(t)).

6, 30, 178, 1246, 9630, 78628, 664918, 5762209, 50849235, 455055615, 4118066401, 37607950281, 346065645810, 3204942065692, 29844571475288, 279238344248557, 2623557165610822, 24739954309690415, 234057667376222382

Offset: 1

Author

Robert G. Wilson v, Oct 30 2000

Keywords

• nonn

Comments

"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)."

Examples

			Li( 10^22 ) = 201467286691248261498.15... => a(22).
pi( 10^22 ) = 201467286689315906290.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• C. Caldwell, values of pi(x)
• B. Riemann, On the Number of Prime Numbers 1859, last page (various transcripts)
• Stephen Wolfram, The Mathematica 3 Book, 1996, Section 3.2.10: Special Functions.

Crossrefs

A052435( 10^n ) = a(n) - pi( 10^n ) for n > 0.

Cf. A000720, A007504.

Programs

Magma

[Round(LogIntegral(10^n)): n in [1..25]]; // G. C. Greubel, May 17 2019

Maple

seq(round(evalf(Li(10^n), 64)), n=1..19); # Peter Luschny, Mar 20 2019

Mathematica

Table[Round[LogIntegral[10^n]], {n, 1, 25}]

PARI

vector(25, n, round(real(-eint1(-log(10^n)))) ) \\ G. C. Greubel, May 17 2019

Sage

[round(li(10^n)) for n in (1..25)] # G. C. Greubel, May 17 2019

Formula

a(n) = round( Li( 10^n )) = round( Ei( log( 10^n ))).