A270857 - OEIS

%I #21 Jul 05 2022 09:53:20

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

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

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

%N Decimal expansion of Sum_{n >= 1} G_n/n^2, where G_n are Gregory's coefficients.

%C Gregory's coefficients (A002206 and A002207) are also known as (reciprocal) logarithmic numbers, Bernoulli numbers of the second kind and Cauchy numbers of the first kind. First few coefficients are G_1=+1/2, G_2=-1/12, G_3=+1/24, G_4=-19/720, etc.

%H G. C. Greubel, <a href="/A270857/b270857.txt">Table of n, a(n) for n = 0..1000</a>

%F Equals integral_{x=0..1} (li(1+x) - gamma - log(x))/x dx, where li(x) is the integral logarithm.

%e 0.4826442216204626123794283911485757739701203962756656...

%p evalf(int((Li(1+x)-gamma-ln(x))/x, x = 0..1), 120);

%t RealDigits[N[Integrate[(LogIntegral[1+x]-EulerGamma-Log[x])/x,{x,0,1}],150]][[1]]

%Y Cf. A270859, A269330, A001620, A002206, A002207, A195189.

%K nonn,cons

%O 0,1

%A _Iaroslav V. Blagouchine_, Mar 24 2016

%E Mathematica program corrected by _Harvey P. Dale_, Jul 05 2022