%I #50 Aug 06 2024 05:06:33
%S 2,3,8,4,2,3,1,0,2,9,0,3,1,3,7,1,7,2,4,1,4,9,8,9,9,2,8,8,6,7,8,3,9,7,
%T 2,3,8,7,7,1,6,1,9,5,1,6,5,0,8,4,3,3,4,5,7,6,9,2,1,0,1,5,0,7,9,8,9,1,
%U 8,1,2,9,3,0,3,6,0,3,7,2,5,5,1,8,6,5,3,5,2,1,0,3,6,5,6,8,0,5,2,0,0,0,2,6,8
%N Decimal expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...
%H G. C. Greubel, <a href="/A079555/b079555.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/pdf/4f03a5e304ec19f8a725774525aecd2a78f4ad81">alternative link</a>.
%F (1/2)*lim sup Product_{k=0..floor(log_2(n)), (1 + 1/floor(n/2^k))} for n-->oo. - _Hieronymus Fischer_, Aug 20 2007
%F (1/2)*lim sup A132369(n)/A098844(n) for n-->oo. - _Hieronymus Fischer_, Aug 20 2007
%F (1/2)*lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - _Hieronymus Fischer_, Aug 20 2007
%F (1/2)*lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - _Hieronymus Fischer_, Aug 20 2007
%F exp(sum{n>0, 2^(-n)*sum{k|n, -(-1)^k/k}})=exp(sum{n>0, A000593(n)/(n*2^n)}). - _Hieronymus Fischer_, Aug 20 2007
%F (1/2)*lim sup A132269(n+1)/A132269(n)=2.3842310290313717241498992886... for n-->oo. - _Hieronymus Fischer_, Aug 20 2007
%F Equals (-1/2; 1/2)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 05 2015
%F 2 + Sum_{k>1} 1/(Product_{i=2..k} (2^i-1)) = 2 + 1/3 + 1/(3*7) + 1/(3*7*15) + 1/(3*7*15*31) + 1/(3*7*15*31*63) + ... (conjecture). - _Werner Schulte_, Dec 22 2016
%F From _Peter Bala_, Dec 15 2020: (Start)
%F The above conjecture of Schulte follows by setting x = 1/2 and t = -1 in the identity Product_{k >= 1} (1 - t*x^k) = Sum_{n >= 0} (-1)^n*x^(n*(n+1)/2)*t^n/( Product_{k = 1..n} 1 - x^k ), due to Euler.
%F Constant C = 1 + Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
%F C = 2 + Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
%F 3*C = 7 + Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
%F 3*7*C = 50 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
%F 3*7*15*C = 751 + Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
%F (End)
%F Equals 1/(1-P), where P is the Pell constant from A141848. - _Gleb Koloskov_, Apr 04 2021
%F Equals Sum_{k>=0} A000009(k)/2^k. - _Vaclav Kotesovec_, Sep 15 2021
%F From _Amiram Eldar_, Feb 19 2022: (Start)
%F Equals (sqrt(2)/2) * exp(log(2)/24 + Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(2))) (McIntosh, 1995).
%F Equals (1/2) * A081845.
%F Equals Sum_{n>=0} 1/A005329(n). (End)
%e 2.38423102903137172414989928867839723877161951650843345769...
%t digits = 105; NProduct[(1 + 1/2^k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 14 2013 *)
%t N[QPochhammer[-1/2,1/2]] (* _G. C. Greubel_, Dec 05 2015 *)
%t 1/N[QPochhammer[1/2, 1/4]] (* _Gleb Koloskov_, Apr 04 2021 *)
%o (PARI) prodinf(n=1,1+2.^-n) \\ _Charles R Greathouse IV_, May 27 2015
%o (PARI) 1/prodinf(n=0, 1-2^(-2*n-1)) \\ _Gleb Koloskov_, Apr 04 2021
%Y Cf. A028362, A048651, A081845, A098844, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132269, A132270, A000593.
%Y Cf. A141848. - _Gleb Koloskov_, Apr 04 2021
%K nonn,cons
%O 1,1
%A _Benoit Cloitre_, Jan 25 2003