A069284 Decimal expansion of li(2) = gamma + log(log(2)) + Sum_{k>=1} log(2)^k / ( k*k! ).

1, 0, 4, 5, 1, 6, 3, 7, 8, 0, 1, 1, 7, 4, 9, 2, 7, 8, 4, 8, 4, 4, 5, 8, 8, 8, 8, 9, 1, 9, 4, 6, 1, 3, 1, 3, 6, 5, 2, 2, 6, 1, 5, 5, 7, 8, 1, 5, 1, 2, 0, 1, 5, 7, 5, 8, 3, 2, 9, 0, 9, 1, 4, 4, 0, 7, 5, 0, 1, 3, 2, 0, 5, 2, 1, 0, 3, 5, 9, 5, 3, 0, 1, 7, 2, 7, 1, 7, 4, 0, 5, 6, 2, 6, 3, 8, 3, 3, 5, 6, 3, 0, 6, 0, 2

Offset: 1

Author

Frank Ellermann, Mar 13 2002

Keywords

• nonn
• cons

Comments

From Mats Granvik, Jun 14 2013: (Start)

The logarithmic integral li(x) = exponential integral Ei(log(x)).

The generating function for tau A000005, the number of divisors of n is: Sum_{n >= 1} a(n) x^n = Sum_{k > 0} x^k/(1 - x^k). Another way to write the generating function for tau A000005 is Sum_{n>=1} A000005(n) x^n = Sum_{a=1..Infinity} Sum_{b>=1} x^(a*b).

If we instead think of the integral with the same form, evaluate at x = exp(1) = 2.7182818284... = A001113 and set the integration limits to zero and sqrt(log(n)), we get for n >= 0:

Logarithmic integral li(n) = Integral_{a = 0..sqrt(log(n))} Integral_{b=0..sqrt(log(n))} exp(1)^(a*b) + EulerGamma + log(log(n)). (End)

li(2)-1 is the minimum [known to date, for n>1] of |li(n) - PrimePi(n)|. - Jean-François Alcover, Jul 10 2013

The modern logarithmic integral function li(x) = Integral_{t=0..x} (1/log(t)) replaced the Li(x) = Integral_{t=2..x} (1/log(t)) which was sometimes used because it avoids the singularity at x=1. This constant is the offset between the two functions: li(2) = li(x) - Li(x) = Integral_{t=0..2} (1/log(t)). - Stanislav Sykora, May 09 2015

Examples

			1.0451637801174927848445888891946131365226155781512015758329...
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 425.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Logarithmic Integral
• Wikipedia, Logarithmic integral function

Crossrefs

Cf. A069285 (continued fraction), A057754, A057794, A060851.

Euler's constant gamma: A001620, log(2): A002162, k*k!: A001563.

Programs

Mathematica

RealDigits[ LogIntegral[2], 10, 105][[1]] (* Robert G. Wilson v, Oct 08 2004 *)

PARI

-real(eint1(-log(2))) \\ Charles R Greathouse IV, May 26 2013

Extensions

Replaced several occurrences of "Li" with "li" in order to enforce current conventions. - Stanislav Sykora, May 09 2015