Little Omega

Wow

Funcion α(n,m)

α(m,n) = α(m^(m)m, n^(n),m^(m), n^(n),m)

α(m,n,0,0,0) = mⁿ

α(m,n,0,1,0) = mⁿ * mⁿ * mⁿ …… mⁿ times

α(m,n,1,0,0) = α(m,n,0,1,0) ^{α(m,n,0,1)^α(m,n,0,1)…α(m,n,0,1) times}

α(m,n,1,1,0) = α(m,n,1,0,0)*α(m,n,1,0)*α(m,n,1,0,0)...α(m,n,1,0) times.

α(m,n,1,2,0) = α(m,n,1,1,0)↑³α(m,n,1,1)

...

α(m,n,o,p,q) = α(α(α(α(m,n,o -1,p - 1)↑^p+1α(m,n,o-1,p -1))))... total q+ times of α(xxx).

Note: if α(n,m) is written merely as α(n), then m will be the same as m.

α(n,m) will convert itself to α(m^{m^m...m times^m}, n^{n^n...n times^n},m^{m^m...m times^m}, n^{n^n...n times^n}, m^{m^m...m times^m}), and α(n) will convert itself to α(n^{n^n...n times^n},n^{n^n...n times^n}, n^{n^n...n times^n}, n^{n^n...n times^n}, n^{n^n...n times^n})

In α(m,n,o,p,q), m,n are the numbers that will be enlarged, while o denotes the number that will be operated, and p tells whether the up arrow notation should enlarge the operation; finally, q means the recursive of α(n) or α(m,n).

Function β(n)

<Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(n^{n^n^n...n times}*3^{2^95^95^95} ) <== Large bracket closes all previous brackets. >

(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(n^{n^n^n...n times}*3^{2^95^95^95} ))

|Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(α(n^{n^n^n...n times}*3^{2^95^95^95} )|

This function is a sizeable recursive sequence of the busy beaver function, fast-growing hierarchy, and the α(n) function from earlier

Function γ(n)

β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(β(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(

Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(Σ(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(ƒ_Γ0(n↑^n↑nn*

2.99081^{95^95^95^95})

An even faster-growing, more considerable function to encapsulate β(n). (With some from the boom def.)

Function ω(n,n₁,n₂,n₃,n₄) (aka little omega)

ω(n) = Rayo^{Rayo(Rayo(10^10^100 * n))}( Rayo(Rayo(10^10^100 x n)))

ω(n,n₁) = α(((ω(n)↑ω⁽ⁿ⁾ω(n)) ↑a) ↑b) ↑C)…….. repeat process n₁ times. )

142443

A

142444443

B

14442444443

C

.

.

.

(A LOT of recursive. Oh also α(n) to top it off.)

ω(n,n₁,n₂) = Tree^{Tree(Ω(n,n1))}(ω(n,n₁))

ω(n,n₁,n₂,n₃) = SCG^{SCG(Ω(n,n1,n2))}(ω(n,n₁,n₂))

ω(n,n₁,n₂,n₃,n₄) = γ(γ(γ(γ(γ(Nucl(SSCG(ω(n,n₁,n₂,n₃))))

The omega function is a function that requires calculating the previous functions. So, ω(1,2,3,4,5) requires calculating ω(1,2,3,4) then ω(1,2,3) then ω(1,2) and finally ω(1).