OFFSET
1,4
COMMENTS
Half the constant A132325.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
FORMULA
Equals (1/2)*lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals (1/2)*lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals (1/2)*lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals (-1/10; 1/10)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015
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
EXAMPLE
1.1122345691370506321260780670944...
MATHEMATICA
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 *)
N[QPochhammer[-1/10, 1/10]] (* G. C. Greubel, Dec 01 2015 *)
PROG
(PARI) prodinf(k=1, 1+1/10^k) \\ Amiram Eldar, May 20 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved