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[]
- ↑ Okojo numbers (Japanese)
- ↑ Old Definition (ver.0 or beta) Archived 2017-11-15. Manually set browzer's encoding to Japanese (EUC).
See also[]
By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
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 Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea
By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system
By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation
By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud
See also: Template:Googology in Asia