An ermine is lecturing the 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)[1], and the current version is 1.1.[2]

"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

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's \(\psi\) · Sushi Kokuu Hen

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