Okojo numbers

The okojo numbers are a small number and its reciprocal, which is a large number. They are created by Japanese googologist Aeton (2013)Okojo numbers (Japanese), and the current version is 1.1.Old Definition (ver.0 or beta) Archived 2017-11-15. Manually set browzer's encoding to Japanese (EUC).
"Okojo" is a Japanese word, which means "ermine" and "stoat" in English, two patterns of names for the same kind of animal. Ermine is an okojo in winter fur, and stoat is an okojo in summer fur. So okojo-ermine number (\(Oe\)) is defined as a small number, and its reciprocal, a large number, is defined as okojo-stoat number (\(Os\)).
In the definition below, the number 54 often appears, because o->0, ko->5 (5 is read as go in Japanese) jo->4 (4 is read as yon, so jo->4 is somewhat forcible though). So anyway, okojo -> 054, and 10 is used in definition of f(n) below, and also 54 is used in f(a,b,...).
\(Oe\) is vastly smaller than googolminex and it is also smaller than 1/Graham's number, but it is of course larger than 0.
==Definition==
*\(f(n)=x\), where \(x\) is the unique real number satisfying \((10\uparrow\uparrow n)^{10\uparrow\uparrow n}=(10\uparrow)^{n+2}x\), where \((10 \uparrow)^k\) for a natural number \(k\) denotes the composite of \(k\) copies of the \(1\)-ary function \(\mathbb{R} \to \mathbb{R} \colon x \mapsto 10^x\).
*\(f(1,1,\square)=f(54,\square)\)
*\(f(1,\square,n)=f(\frac{1}{f(1,\square,n-1)},\square)\), here \(\frac{1}{f(1,\square,n-1)}\) might not be integer, so when substituting, round it off (and so forth).
*\(f(\blacksquare,m,1,\square)=f(\blacksquare,m-1,54,\square)\)
*\(f(\blacksquare,m,\square,n)=f(\blacksquare,m-1,\frac{1}{f(\blacksquare,m,\square,n-1)},\square)\)
Where:
*\(\square\): vector of 1, with the length larger than or equal to 0
*\(\blacksquare\): vector of integers larger than or equal to 1, with the length larger than or equal to 0
*\(m>1\), and \(n>1\)
Using this function, the okojo numbers are defined as:
*\(Oe(n)=f(\underbrace{1,1,\dots,1}_{n\text{ copies of }1},1)\)
*\(Oe(54)\) = Okojo-ermine Number (\(Oe\))
*\(\frac{1}{Oe}\) = Okojo-stoat Number (\(Os\)), \(\frac{1}{Oe(n)}=Os(n)\)
\(Os\) and \(Os(n)\) might not be integer, but they don't need to be rounded off.
== Approximations ==
Notation 	Approximation for Oe 	Approximation for Os
BEAF 	\(\{54,55(1)2\}^{-1}\) 	\(\{54,55(1)2\}\)
Cascading-E notation 	N/A 	E54#^#54
Fast-growing hierarchy	\(f_{\omega^\omega}(53)^{-1}\)	\(f_{\omega^\omega}(53)\)
N-growing hierarchy 	
\([54]_{\omega^\omega}(53)^{-1}\approx N_{\omega^\omega}(53)^{-1}\)	\([54]_{\omega^\omega}(53)\approx N_{\omega^\omega}(53)\)
== Sources ==
==See also==
ja:おこじょ数
Category:Numbers
Category:Googology in Asia
Category:Suspected transcendental numbers
Category:Animal-based name
Category:Polynomial omega level
Category:Micronyms