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%I #16 May 20 2023 07:03:16
%S 1,1,1,2,2,3,4,5,6,9,1,3,7,0,5,0,6,3,2,1,2,6,0,7,8,0,6,7,0,9,4,4,0,5,
%T 8,0,3,7,4,7,5,0,7,4,6,7,5,7,7,5,9,2,8,3,5,7,8,7,9,5,8,2,3,7,0,3,3,2,
%U 5,3,4,6,9,4,8,8,1,4,1,1,0,4,3,7,6,4,7,2,2,2,2,6,4,2,1,3,5,2,3,5,5,6,4,7,4
%N Decimal expansion of Product_{k>=1} (1+1/10^k).
%C Half the constant A132325.
%H G. C. Greubel, <a href="/A132326/b132326.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 Equals (1/2)*lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
%F Equals (1/2)*lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
%F Equals (1/2)*lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
%F Equals exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*10^n)).
%F Equals (1/2)*lim sup_{n->oo} A132271(n+1)/A132271(n).
%F Equals (-1/10; 1/10)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 01 2015
%F Equals (sqrt(2)/2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - _Amiram Eldar_, May 20 2023
%e 1.1122345691370506321260780670944...
%t digits = 105; NProduct[1+1/3^k, {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t N[QPochhammer[-1/10,1/10]] (* _G. C. Greubel_, Dec 01 2015 *)
%o (PARI) prodinf(k=1, 1+1/10^k) \\ _Amiram Eldar_, May 20 2023
%Y Cf. A000593, A067080, A079555, A132019-A132026, A132034-A132038, A132265-A132268, A132271, A132272, A132325.
%K nonn,cons
%O 1,4
%A _Hieronymus Fischer_, Aug 20 2007
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