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This is a function that use an algorithm produces fairly big numbers. 

#(n)

To obtain the output : choose a integer n superior to 1 and follow the first step.

Step a (for a<n) : Repeat n times step a+1

Step n : Set n to n+1. 

The end result is written as #(n). This may seem weak, but if we just change step n to : ”Repeat n times step 1.0” and define step 1.0 as “Set n to n+1”, we already have something faster than the graham function ! . Its just f_ω(n+1). Now we can define #_1(n) and #_x(n).

#_1(n)

Step a (for a<n) : Repeat n times step a+1

Step n : Repeat n times step 1.0

Step a.b : (for a and b <n)

Repeat n times step a.b+1 Step a.n (for a<n): Repeat n times step a+1.0

Step n.0 : Add 1 to n

#_x(n) is an extension of #_1(n) that goes up to step n.0.0.0.0.0.0… with x zeroes.

The Supreme Algorithmic Function is #_n(n) and has a growth rate of f_(ω^ω) f_ω(n^n) using FGH.

#_Δ(n)

Δ_1(x) is step x.0.0.0.0... with x terms.

You can think of #(n) as simply a column of steps, #_1(n) as a collection of any columns : a plane of steps, #_2(n) a cube of steps, and #_x(n) as a (x+1)-cube.

Now, you can imagine step Δ_1(x) as x x-cubes, Δ_1(x,0) as x columns of x-cubes, Δ_1(x,0,0) as x cubes of x-cubes (x cubes = x 3 dimensional cubes; x-cubes = some number of x-dimensional cubes) , etc.

Δ_2(x) is x x-cubes of x-cubes.

Δ_3(x) is x x-cubes of x-cubes of x-cubes.

Etc.

The Supreme Metacube Function (#_Δ(n)) is a function that uses the algorithm up to Δ_n(n) and has a growth rate of f_(epsilon(0)) f_w(n^^n)

#_ω(n)

We can still define steps that come after !

ω_1(n) is n columns/lines of "n-cube of n-cubes of n-cubes of n-cubes..."

ω_1(n,0) is n planes

ω_1(n,0,0) is n cubes

etc.

ω_2(x) is n n-cubes of n-cubes.

ω_3(x) is n n-cubes of n-cubes of n-cubes.

Etc.

The Supreme Metametacube Function (#_ω(n)) is a function that uses the algorithm up to ω_n(n).

Growth : f_w(n^^n^^n)

#_Ω(n)

You can probably see the pattern between #_Δ(n) and #_ω(n). We can continue the pattern and use the hypercube thingy in the index itself !! This gets us to the last extension of the function (for now) : #_Ω(n). Basically : its a metametametameta... (the "meta" form a metametametameta...(the "meta" form a metametametameta...(the "meta" form a metametametameta...(the "meta" form a metametametameta...(...cube)cube)cube)cube)...cube.

I call it : the hypermeta-cube function

Growth rate : f_w(n^^^n)

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