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A069284 Decimal expansion of Li(2) = gamma + log(log(2)) + sum_{k=1..inf} log(2)^k / ( k*k! ). 1
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

Li(x) = Li(2) + integral 1/log(t) dt from 2 to x > 2.

Contribution 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) = Integrate_{a = 0..Sqrt(Log(n))} Integrate_{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]

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

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, ln(2): A002162, k*k!: A001563.

Sequence in context: A131131 A073241 A094642 * A068447 A237109 A199384

Adjacent sequences:  A069281 A069282 A069283 * A069285 A069286 A069287

KEYWORD

nonn,cons

AUTHOR

Frank Ellermann, Mar 13 2002

STATUS

approved

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Last modified November 24 16:46 EST 2014. Contains 249899 sequences.