Fish number 4 (F4) is a number defined by Japanese googologist Fish in 2002. It is the smallest of the Fish numbers that is defined using an uncomputable function.

s'(1) map is a function which maps functions to functions, as follows.

Function $$s'(1)f$$ is a busy beaver function for an oracle machine having an oracle which calculates function $$f$$. That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is $$s'(1)f(n)$$.

By comparing with the order-n busy beaver function $$\Sigma_n(x)$$, let $$f$$ be a computable function. Then it's easy to see that (exponents mean iteration of the map here):

\begin{eqnarray*} s'(1)f & = & \Sigma_1(x)\\ s'(1)^2f & = & \Sigma_2(x)\\ s'(1)^3f & = & \Sigma_3(x)\\ s'(1)^nf & = & \Sigma_n(x)\\ s'(1)^xf & = & \Sigma_x(x)\end{eqnarray*}

For $$n>1$$, $$s'(n)$$ map is defined similar to the s(n) map,

\begin{eqnarray*} s'(n)f & = & s'(n-1)^{x}f(x) (\text{for } n>1) \\ \end{eqnarray*}

After this, the definition is similar to Fish number 3;

\begin{eqnarray*} ssʹ(1)f & = & sʹ(x)f(x) \\ ssʹ(n)f & = & [ssʹ(n − 1)^{x}]f(x) (\text{for } n>1) \\ F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\ F_4 & = & F_4^{63}(3) \end{eqnarray*}

## Sources

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's $$\psi$$
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
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 computation programmes · TR function (I0 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 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
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 computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: 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 · YY sequence · Y function · ω-Y sequence