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%I #15 Feb 19 2022 07:08:00
%S 1,5,6,4,9,3,4,0,1,8,5,6,7,0,1,1,5,3,7,9,3,8,8,4,9,1,0,6,7,2,8,8,3,5,
%T 4,1,6,5,6,9,4,2,5,9,1,9,8,9,5,0,3,5,0,0,9,4,9,6,7,2,1,0,2,9,9,2,3,0,
%U 2,1,1,0,7,2,5,8,0,9,6,7,6,6,9,3,9,0,3,6,6,0,3,6,7,7,2,9,6,3,8,8,1,5,2,6,0
%N Decimal expansion of Product_{k>=1} (1+1/3^k).
%C Half the constant A132323.
%H G. C. Greubel, <a href="/A132324/b132324.txt">Table of n, a(n) for n = 1..1200</a>
%H Richard J. McIntosh, <a href="https://doi.org/10.1112/jlms/51.1.120">Some Asymptotic Formulae for q-Hypergeometric Series</a>, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.993.6895&rep=rep1&type=pdf">alternative link</a>.
%F (1/2)*lim sup Product{k=0..floor(log_3(n)} (1+1/floor(n/3^k)) for n-->oo.
%F (1/2)*lim sup A132327(n)/A132027(n) for n-->oo.
%F (1/2)*lim sup A132327(n)/n^((1+log_3(n))/2) for n-->oo.
%F (1/2)*lim sup A132328(n)/n^((log_3(n)-1)/2) for n-->oo.
%F exp(Sum_{n>0} 3^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*3^n)).
%F (1/2)*lim sup A132327(n+1)/A132327(n) = 1.56493401856701153793884910... for n-->oo.
%F Equals (-1/3; 1/3)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 01 2015
%F From _Amiram Eldar_, Feb 19 2022: (Start)
%F Equals (sqrt(2)/2) * exp(log(3)/24 + Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(3))) (McIntosh, 1995).
%F Equals Sum_{n>=0} 1/A027871(n). (End)
%e 1.56493401856701153793884910...
%t digits = 105; NProduct[1+1/3^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t N[QPochhammer[-1/3,1/3]] (* _G. C. Greubel_, Dec 01 2015 *)
%Y Cf. A027871, A079555, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132327, A132328, A000593.
%K nonn,cons
%O 1,2
%A _Hieronymus Fischer_, Aug 20 2007
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