I just figured I would sign up because this looks interesting. I am not an expert mathematician or anything, but I do have some Googology-related ideas.

First is what I call the sum of 3 cubes function, which I defined anonymously in a forum post but no one responded. Here it is again:

So I was watching this video:

And I came up with this idea.

For all X which meet the following conditions:

  • There exist integers a, b, and c such that a^3 + b^3 + c^3 = X
  • a, b, and c can include repeats of the same integer (a can equal b can equal c)
  • a, b, and c can be positive or negative
  • At least one of a, b, and c is > X

Define FX(n) as the nth smallest positive value of a^3, b^3, or c^3 (whichever is larger) such that a^3 + b^3 + c^3 = X

So for example, using some of the numbers they addressed in the video:

F29(1) = 27

F30(1) = 2,220,422,932^3

So we have a family of functions that grow at quite different rates.

Now define Y as the value of X for which FX(n) grows fastest where Y <= Z, and Z >= 1

So now we have the function FY(Z), which gives an X. For example, FY(100) would give you the integer X <= 100 for which FX(n) grows fastest.

Now define F(Z, n) = FX(n) where X = FY(Z)

My second idea is simpler. I call it the Go function. Go(n) = the maximum amount of distinct games of Go that can be played on a board with size n by n lines.

My third idea is what I call a Conway factorial (C!)

C!(n) = n -> (n-1) -> (n-2)... etc. until you get to 1, using Conway chained arrow notation.

My fourth idea is the one I am most unsure about. It might be a naive extension or whatever, but basically the idea is to take the Rayo hierarchy as defined in the page for Fish Number 7 and create the function RRR, where RRR(n) equals the smallest positive integer bigger than any finite positive integer that can be defined in the Rayo hierarchy using n symbols or less.

Are any of these good ideas?

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