User:Cloudy176/Department of bubbly negative numberbottles/Mixed factorial: The lost extensions


 * Content below was copied from this revision of Mixed factorial; see that page's history for attribution
 * See also: Talk:Mixed factorial

Mixed factorial \(n^*\) is a function recursively defined as

\[1^* = 1\]

\[(n + 1)^* = n^* +^n (n + 1)\]

where \(+^n\) is the \(n\)th hyper operator, starting at addition. For example, \(4^* = ((1 + 2) \cdot 3) \uparrow 4\). Informally, the sequence can be visualized as starting with 1, adding 2, multiplying by 3, exponentiating by 4, tetrating by 5, ...

The function was coined by an author under the alias of "".

Extended Mixed Factorials
By repeating the process of taking mixed factorial we get numbers of form (n*)*, ((n*)*)* etc. or, equivalently, n**, n***, n****... These can be also written as n*2, n*3, etc.

First Extension
For even farther extension, we can say n*x* ... x , with k x's can be iterated as n*(x,k). As you can see, x represents the mixed factorial numbers, and k represents the amount of them.


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Second Extension
Even farther extension can represented in this diagram:

As you can see, 2 arrays are represented by n*[k,x:2], 3 arrays are represented by n*[k,x:3], 4 arrays are represented by n*[k,x:4], and so on.

Third Extension
Farther extensions can be represented in the examples below: n*[k,x#5], n*[k,x#6], etc. If this isn't very clear to you, get a piece of paper and try working this out. It's a bit complicated, but makes huge numbers.
 * n*[k,x:(n* [k,x])] = n*[k,x#2]
 * n*[k,x:(n* [k,x:(n* [k,x])] )] = n*[k,x#3]
 * n*[k,x:(n* [k,x:(n* [k,x:(n* [k,x])] )] )] = n*[k,x#4]

Fourth Extension
The fourth extension can be represented in the examples below. As said before, try working it out on paper if you don't get it. It has multiple #'s. n*[k,x##5], n*[k,x##6], n*[k,x##7], etc.
 * n*[k,x#(n* [k,x])] = n*[k,x##2] or n*[k,x#2
 * n*[k,x#(n* [k,x#(n* [k,x])] )] = n*[k,x##3] or n*[k,x#2
 * n*[k,x#(n* [k,x#(n* [k,x#(n* [k,x])] )] )] = n*[k,x##4] or n*[k,x#2

For three #'s:
 * n*[k,x##(n* [k,x])] = n*[k,x###2] or n*[k,x#3
 * n*[k,x##(n* [k,x##(n* [k,x])] )] = n*[k,x###3] or n*[k,x#3

Four #'s:
 * n*[k,x###(n* [k,x])] = n*[k,x####2] or n*[k,x#4
 * n*[k,x###(n* [k,x###(n* [k,x])] )] = n*[k,x####3] or n*[k,x#4

Five #'s: Six #'s, seven #'s, and so on. Note: As for the | symbol, it is just a seperator. If it wasn't there, it would look like pentation instead of the amount of #'s.
 * n*[k,x####(n* [k,x])] = n*[k,x#####2] or n*[k,x#5
 * n*[k,x####(n* [k,x####(n* [k,x])] )] = n*[k,x#####3] or n*[k,x#5

Fifth Extension

 * n*[k,x#(n* [k,x#)|n] = n*[k,x≍ or n*[k,x#1,2
 * n*[k,x#(n* [k,x# (n* [k,x#)|n] ) |y] = n*[k,x≍ or n*[k,x#1,2

We have 2 ≍'s, 3 ≍'s, etc.

On 3 in 2nd entry.
 * n*[k,x≍(n* [k,x≍)|n] = n*[k,x#1,3
 * n*[k,x≍(n* [k,x≍ (n* [k,x≍)|n] ) |n] = n*[k,x#1,3

On 4 in 2nd entry.
 * n*[k,x#(n* [k,x≍ 1,3),3 |n] = n*[k,x#1,4
 * n*[k,x#(n* [k,x≍ (n* [k,x≍ 1,3),3 |n] ),3 |n] = n*[k,x#1,4

We extend to 3 entries:
 * n*[k,x#1,(n* [k,x#)|n] = n*[k,x#1,2
 * n*[k,x#1,(n* [k,x# 1,(n* [k,x#)|n] ) |y] = n*[k,x#1,1,2
 * n*[k,x#1,(n* [k,x# 1,1,2),2 |n] = n*[k,x#1,1,3
 * n*[k,x#1,(n* [k,x# 1,(n* [k,x# 1,1,2),2 |n] ),2 |y] = n*[k,x#1,1,3

And the four entries, five entries, etc.

Sixth Extension
Iterating large amounts of entries can be explained in the following: Let's use q as the amount of entries, and s as the number in the entries, we could iterate that as '''n*[k,x#s,s,s,s. . . s,s,s,s (with q entries) as n*[k,x#s[2]q'''.

We defined n*[k,x#s[2]1,2 = n*[k,x#s[2]n* [k,x# s[2]n* [k,x# s |3] |3]  and n*[k,x#s[2]q[2]q 2 = n*[k,x#s[2]q,q...q,q (q2 q's)

Also n*[k,x#s[w+1]q = n*[k,x#s[w]s...s[w]s (q s's).

Also we have nested arrays. For example: n*[k,x#s[1,2]q = n*[k,x#s[n* [k,x# s[n* [k,x# s,q]q |3] ]q |3] .

Seventh Extension

 * n*[k,x#n[n, 2m-1]n,2m-1 = n*[k,x#1, 2m-1
 * n*[k,x#n[n[n, 2m-1]n,2m-1]n,2m-1 = n*[k,x#1, 2m

Other arrays work same.


 * n*[k,x#n[n, om-1]o-1n,om-1 = n*[k,x#1, om-1
 * n*[k,x#n[n[n, om-1]o-1n,om-1]o-1n,om-1 = n*[k,x#1, om-1

Eighth Extension
...There

Ninth Extension
...is

Tenth Extension
...no

Eleventh Extension
...extension

Don't uncomment this. It you do, i will comment it.
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Some numbers

 * Tetrafact, 4*
 * Pentafact, 5*
 * Hexafact, 6*
 * Centifact, 100*
 * Double Centifact, 100** = 100*2 (100*2)
 * Triple Centifact, 100*** = 100*3 (100*3)
 * Quadruple Centifact, 100**** = 100*4 (100*4)
 * Quintuple Centifact, 100***** = 100*5 (100*5)
 * Dectuple Centifact, 100*10 (100*10)
 * Centuple Centifact, 100*100 (100*100)
 * Duo-centifact, 100*[100,2] (100*(100,2))
 * Hypercentifact, 100*[100,100] (100*(100,100))
 * Duo-hypercentifact, 100*[100,100:2] (100*(100,100:2))
 * Ultracentifact, 100*[100,100:100] (100*(100,100:100))
 * Duo-ultracentifact, 100*[100,100#2]
 * Kilocentifact, 100*[100,100#100]
 * Duo-kilocentifact, 100*[100,100#2
 * Megacentifact, 100*[100,100#100
 * Duo-megacentifact, 100*[100,100#100,100
 * Gigacentifact, 100*[100,100#100[2]100undefined
 * Duo-gigacentifact, 100*[100,100#100[100[2]100]100undefined
 * Teracentifact, 100*[100,100#100[...[100]...]100undefined (100 nested)
 * Duo-teracentifact, 100*[100,100#100, 100[...[100]...]100100undefined (100 nested)
 * Petacentifact, 100*[100,100#100, ... 100... 100undefined (100 nested)