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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 (list; constant; graph; refs; listen; history; text; internal format)
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)|. - Jean-Fran├žois 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

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Last modified December 4 13:19 EST 2016. Contains 278750 sequences.