Facthype Notation

Facthype notationArcioko, Facthype Notation, Google site. Retrieved at UTC 2:00 Janary 29th 2022. (FHN), short for Factorial Hyperoperator Notation, is a notation created by googology wiki user Arcioko, based on the factorial function. It is currently in development.Arcioko, Notations page, Google site. Retrieved at 19:40 UTC January 29th 2022
==Basic FHN==
According to Arcioko, every variable in the definition runs through the set of natural numbers, and perhaps "natural number" in this context means "positive integer". The factorial hyperoperation function is an extension to the Nested Factorial Notation, it is a vital part of FHN:
*\(n!^{1,m} = n!^m\)
*\(n!^{k,1} = n!\)
*\(n!^{k,m} = n!^{k-1,n!^{k,m-1}}\)
==Rules==
The definition on the website says:
*\(n(1)\#^{k,m} = n!^{k,m}\)
*\(n(h)\#_{a_1,a_2, \cdots, a_{i-1}, a_i, 1} = n(h)\#_{a_1,a_2, \cdots, a_{i-1}, a_i}\)
*\(n(h)\#_{a_1,a_2, \cdots, a_{i-1}, a_i} = \)
**Let \(A_n = n + \sum_{k=1}^{i-s} a_{s-k+1}\)
**\(n(h)\#_{a_1,a_2, \cdots, a_{i-1}, a_i} = n(h)\#_{A_1,A_2, \cdots , A_{i-1}, a_i - 1}\)
*\(n(h+1)\#_1\ = n(h)\#^{n,n}_{\underbrace{n,n,\cdots,n,n}_{h-1}}\)
*\(n(h)\#^m_{a_1, a_2, \cdots , a_{i-1}, a_i} = \)
**\(n(h)\#^1_{a_1, a_2, \cdots , a_{i-1}, a_i} = n(h)\#_{a_1,a_2, \cdots, a_{i-1}, a_i}\)
**\(n(h)\#^{m}_{a_1, a_2, \cdots , a_{i-1}, a_i} = (n(h)\#^{m-1}_{a_1, a_2, \cdots , a_{i-1}, a_i})(h)\#_{a_1, a_2, \cdots , a_{i-1}, a_i}\)
*\(n(h)\#^{k,m}_{a_1, a_2, \cdots , a_{i-1}, a_i}\)
**\(n(h)\#^{1,m}_{a_1, a_2, \cdots , a_{i-1}, a_i} = n(h)\#^m_{a_1, a_2, \cdots , a_{i-1}, a_i}\)
**\(n(h)\#^{k,1}_{a_1, a_2, \cdots , a_{i-1}, a_i} = n(h)\#_{a_1, a_2, \cdots , a_{i-1}, a_i}\)
**\(n(h)\#^{k,m}_{a_1, a_2, \cdots , a_{i-1}, a_i} = n(h)\#^{k-1,n(h)\#^{k,m-1}_{a_1, a_2, \cdots , a_{i-1}, a_i}}_{a_1, a_2, \cdots, a_{i-1}, a_i}\)
The "more stuff" section defines \(n!\ //\ m\) as \(n(m+1)\#_1\), Arcioko intends \(n!\ //\ 3\) to have a growth rate of \(f_{\omega^\omega}(n)\) and \(n!\ //\ 4\) to have a growth rate of \(f_{\epsilon_0}(n)\). According to Arcioko, the notation is to be extended. Arcioko also made the function \(n\ //\ ! = n!\ //\ n\) which according to arcioko "i dont know how fast it grows but it probably grows fast as hell" and coined it as "superfactorial." According to arcioko this was "to assert dominance in pickover and sloane."
The following phrase is cited from the original source:
2Slow4me, as some corporation trying to relate to a younger audience would say
- Arcioko
==Sources==
Category:Incomplete
Category:Factorials
Category:Notations