|
%I #12 May 09 2023 08:56:49
%S 8,3,6,7,9,5,4,0,7,0,8,9,0,3,7,8,7,1,0,2,6,7,2,9,7,9,8,1,4,6,1,3,6,2,
%T 4,1,3,5,2,4,3,6,4,3,5,8,7,6,7,1,6,5,1,9,9,6,4,1,1,5,1,0,1,7,7,0,0,9,
%U 1,6,0,1,2,6,5,4,2,7,6,0,5,8,7,8,7,5,5,5,4,2,8,4,9,0,5,1,2,0,2,1,7,5,3
%N Decimal expansion of Product_{k>0} (1-1/7^k).
%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 Equals exp(-Sum_{n>0} sigma_1(n)/(n*7^n) = exp(-Sum_{n>0} A000203(n)/(n*7^n)).
%F From _Amiram Eldar_, May 09 2023: (Start)
%F Equals sqrt(2*Pi/log(7)) * exp(log(7)/24 - Pi^2/(6*log(7))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(7))) (McIntosh, 1995).
%F Equals Sum_{n>=0} (-1)^n/A027875(n). (End)
%e 0.8367954070890378710...
%t digits = 103; NProduct[1-1/7^k, {k, 1, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t RealDigits[QPochhammer[1/7], 10, 100][[1]] (* _Amiram Eldar_, May 09 2023 *)
%o (PARI) prodinf(k=1, 1 - 1/(7^k)) \\ _Amiram Eldar_, May 09 2023
%Y Cf. A027875, A048651, A100220, A132023, A132026, A132038, A098844, A067080, A000203.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 14 2007
|
