A388042 - OEIS

%I #116 Mar 07 2026 18:24:31

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

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

%U 3,6,2,8,9,4,8,8,8,6,2,4,2,9,2,2,6,1,4,2,3,3,0,9,2,0,8,1,2,0,7,7,4,4,2,9,4

%N Decimal expansion of Integral_{x>=e} log(log(x))/(1+x^2) dx.

%C Integral_{x=0..1/e} log(log(x))/(1+x^2) is complex number 0.2152489028... + i*1.107453576148368... = A388042 + i*A393630.

%C Integral_{x=1/e..1} log(log(x))/(1+x^2) is complex number -0.475691709... + i*1.35994752412397... = -A393691 + i*A390125.

%C Integral_{x=0..1} log(log(x))/(1+x^2) is complex number -0.2604428063... + i*2.467401100272... = A115252 + i*A091476.

%C Integral_{x>=0} log(log(x))/(1+x^2) is complex number -0.5208851637565... + i*2.467401100272... = -2*A115252 + i*A091476.

%C Integral_{x=1..e} log(log(x))/(1+x^2) is real negative number -0.47569170911830734... = -A393691.

%C Integral_{x>=e} log(log(x))/(1+x^2) is real positive number 0.215248902817318895... = A388042.

%H Sean A. Irvine, <a href="/A388042/b388042.txt">Table of n, a(n) for n = 0..999</a>

%F Equals real part of Integral_{x=0..1/e} log(log(x))/(1+x^2).

%F Equals Integral_{x=arctan(e)..Pi/2} log(log(tan(x))) dx.

%F Equals Integral_{x>=1} log(x)/(2 * cosh(x)) dx.

%F Let A = 0.215248902817318895... B = -0.47569170911830734... then

%F A+B = (Pi/4) log(4*Pi^3/Gamma(1/4)^4) = -0.260442806300988...

%F Equals Sum_{k>=0} (-1)^(k+1) * Ei(-(2*k+1)) / (2*k+1). - _Sean A. Irvine_, Mar 04 2026

%e 0.21524890281731889512316472...

%p evalf(Int(log(log(x))/(1+x^2), x = exp(1)..infinity), 105); # _Vaclav Kotesovec_, Feb 24 2026

%t RealDigits[NIntegrate[Log[Log[x]]/(1 + x^2), {x, E, Infinity}, WorkingPrecision -> 110], 10, 105][[1]]

%t (* or *)

%t NSum[(-1)^(k + 1) ExpIntegralEi[-(2*k + 1)]/(2*k + 1), {k, 0, Infinity}, WorkingPrecision -> 105]

%Y Cf. A091476, A115252, A390125, A393630, A393691.

%K nonn,cons

%O 0,1

%A _Artur Jasinski_, Feb 22 2026