

A069284


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


2



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
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OFFSET

1,3


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..Infinity} A000005(n) x^n = Sum_{a=1..Infinity} Sum_{b=1..Infinity} 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).  JeanFrançois 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: log(2)=li(x)Li(x)=integral[t=0..2](1/log(t)).  Stanislav Sykora, May 09 2015


REFERENCES

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


LINKS

Table of n, a(n) for n=1..105.
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function


EXAMPLE

1.0451637801174927848445888891946131365226155781512015758329...


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A069285 (continued fraction), A057754, A057794, A060851.
Euler's constant gamma: A001620, log(2): A002162, k*k!: A001563.
Sequence in context: A131131 A073241 A094642 * A272638 A068447 A237109
Adjacent sequences: A069281 A069282 A069283 * A069285 A069286 A069287


KEYWORD

nonn,cons


AUTHOR

Frank Ellermann, Mar 13 2002


EXTENSIONS

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


STATUS

approved



