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

%I A069284 #67 Feb 16 2025 08:32:46
%S A069284 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,
%T A069284 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,
%U A069284 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
%N A069284 Decimal expansion of li(2) = gamma + log(log(2)) + Sum_{k>=1} log(2)^k / ( k*k! ).
%C A069284 From _Mats Granvik_, Jun 14 2013: (Start)
%C A069284 The logarithmic integral li(x) = exponential integral Ei(log(x)).
%C A069284 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).
%C A069284 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:
%C A069284 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)
%C A069284 li(2)-1 is the minimum [known to date, for n>1] of |li(n) - PrimePi(n)|. - _Jean-François Alcover_, Jul 10 2013
%C A069284 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
%D A069284 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 425.
%H A069284 Vincenzo Librandi, <a href="/A069284/b069284.txt">Table of n, a(n) for n = 1..5000</a>
%H A069284 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A069284 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>
%e A069284 1.0451637801174927848445888891946131365226155781512015758329...
%t A069284 RealDigits[ LogIntegral[2], 10, 105][[1]] (* _Robert G. Wilson v_, Oct 08 2004 *)
%o A069284 (PARI) -real(eint1(-log(2))) \\ _Charles R Greathouse IV_, May 26 2013
%Y A069284 Cf. A069285 (continued fraction), A057754, A057794, A060851.
%Y A069284 Euler's constant gamma: A001620, log(2): A002162, k*k!: A001563.
%K A069284 nonn,cons
%O A069284 1,3
%A A069284 _Frank Ellermann_, Mar 13 2002
%E A069284 Replaced several occurrences of "Li" with "li" in order to enforce current conventions. - _Stanislav Sykora_, May 09 2015