Okojo numbers

Okojo Numbers are a couple of small number and its reciprocal, which is large number. They are created by Japanese googologist Aeton (2013), and now the version is 1.1.

"Okojo" is a Japanese word, which means, in English, ermine and stoat, 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 definded as a small number, and its reciprocal large number is definded as okojo-stoat number\(Os\).

In the definition below, 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,...).

Definition

 * \(f(n)=x\), when \((10\uparrow\uparrow n)^{10\uparrow\uparrow n}=(10\uparrow)^{n+2}x\)


 * \(f(1,1,\square)=f(54,\square)\)
 * \(f(1,\square,n)=f(\frac{1}{f(1,\square,n-1)},\square)\), here \(\frac{1}{f(1,\square,n-1)}\) might not be integer, so when substituting, round it off (and so forth).
 * \(f(\blacksquare,m,1,\square)=f(\blacksquare,m-1,54,\square)\)
 * \(f(\blacksquare,m,\square,n)=f(\blacksquare,m-1,\frac{1}{f(\blacksquare,m,\square,n-1)},\square)\)

when,
 * \(\square\): 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
 * \(m,n\): an integer more larger than 1


 * \(Oe(n)=f(\underbrace{1,1,\dots,1}_{n\text{ copies of }1},1)\)

\(Os\) and \(Os(n)\) might not be integer, but it doesn't need to be rounded off.
 * \(Oe(54)\) = Okojo-ermine Number (\(Oe\))
 * \(\frac{1}{Oe}\) = Okojo-stoat Number (\(Os\)), \(\frac{1}{Oe(n)}=Os(n)\)