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A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

%I A132034 #28 Aug 06 2024 05:04:04
%S A132034 8,0,5,6,8,7,7,2,8,1,6,2,1,6,4,9,4,0,9,2,3,7,5,0,2,1,5,4,9,6,2,9,8,9,
%T A132034 6,8,9,1,7,9,9,7,6,2,8,6,9,3,9,2,6,6,9,2,0,8,5,7,5,6,8,1,0,0,7,2,1,9,
%U A132034 4,1,0,5,4,8,2,0,3,6,2,0,2,0,4,5,6,3,0,4,3,7,7,0,0,5,3,2,8,0,2,7,5,2,1
%N A132034 Decimal expansion of Product_{k>0} (1-1/6^k).
%H A132034 G. C. Greubel, <a href="/A132034/b132034.txt">Table of n, a(n) for n = 0..1200</a>
%H A132034 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/pdf/4f03a5e304ec19f8a725774525aecd2a78f4ad81">alternative link</a>.
%F A132034 Equals exp( -Sum_{n>0} sigma_1(n)/(n*6^n) ).
%F A132034 Equals (1/6; 1/6)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Nov 30 2015
%F A132034 From _Amiram Eldar_, May 09 2023: (Start)
%F A132034 Equals sqrt(2*Pi/log(6)) * exp(log(6)/24 - Pi^2/(6*log(6))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(6))) (McIntosh, 1995).
%F A132034 Equals Sum_{n>=0} (-1)^n/A027873(n). (End)
%e A132034 0.805687728162164940923750...
%t A132034 digits = 103; NProduct[1-1/6^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+20] // N[#, digits+20]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t A132034 N[QPochhammer[1/6,1/6]] (* _G. C. Greubel_, Nov 30 2015 *)
%o A132034 (PARI) prodinf(x=1, 1-(1/6)^x) \\ _Altug Alkan_, Dec 01 2015
%Y A132034 Cf. A000203, A027873, A048651, A067080, A098844, A100220, A132022, A132026, A132038.
%K A132034 nonn,cons
%O A132034 0,1
%A A132034 _Hieronymus Fischer_, Aug 14 2007