Related to N-ation of N by N
I would like to know of a function SFG(x) with following properties:
- Grows at least as fast as Ackermann sequence
- Defined for all reals greater than 1
- Has inverse function defined for all reals after 1
- Smooth for all parameter values after 1
- Easy to compute for commonly used numbers (e.g. Planck volume to approximate observable universe volume in Planck volumes)
- Supported with mathematical apparatus to make it possible to compare SFG(x) with a given non-smooth fast growing function for a given x
Commonly cited fast growing functions are Busy Beaver numbers and TREE, but they fail almost all of bullet points above. Once there is a sufficiently fast growing function, all other fast growing functions' known values can be approximated as SFG(x), where x is relatively small real number. Then math can be made in SFG-notation instead of natural numbers-based notation.