Factorial Types

In this page, I will describe the different factorials there are, and also some of my factorial functions I made.

I'm sure you know what factorials are, but if you have been living under a rock for a while, here's what it means:

"The factorial is a function applicable to any non-negative integer n" Source

Alright, enough explaining, lets get into all of the factorials!

Lets start with the factorial I'm sure a lot of you know by now, n!

n! = n x (n - 1) x (n - 2) x (n - 3) x .... x 4 x 3 x 2 x 1

First 10 values of n!

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5040

8! = 40320

9! = 362880

10! = 3628800

I have already done 3 versions of a video I did with factorials, so go watch that if your interested more.

The next one is... Multifactorial

So, 3!! ≠ 3! x 2! x 1!, but the number of exclamation marks there are actually effects the outcome, which makes multifactorial slower than normal factorial.

Example:

n!! = n x (n - 2) x (n - 4) x .... x 6 x 4 x 2

6!! = 6 x 4 x 2 = 48

[Note]

That example only works for even numbers. For odd numbers it would be...

n!! = n x (n - 2) x (n - 4) x .... x 5 x 3 x 1

7!! = 7 x 5 x 3 x 1 = 105

Now for the next one, which is more powerful than normal factorials, and actually has three different definitions. This one is called "Superfactorial", and uses $ instead of !.

First definiton: Pickover

n$ = n! ↑↑ n! = n! ↑↑↑ 2

You can imagine just how large this thing can get in only a few entries...

1$ = 1

2$ = 4

3$ = 10^10^10^10^36305.315801918918...

4$ = E1.52#24

It's extremely big after 3$, so large you wouldn't even be able to calculate all of that.

Second definition: Sloane and Plouffe

Same symbol used

n$ = n! x (n - 1)! x (n - 2)! x .... x 3! x 2! x 1!

Not as powerful as the one mentioned eariler, but is more controllable.

1$ = 1

2$ = 2

3$ = 12

4$ = 288

5$ = 34560

Third definition: Daniel Corrêa

Again, uses the same symbol.

n$ = (11...11n) x ((11...11n)!) x ((11...11n)!^2) ... ((111n)!^(n-3) x ((111n)!^(n-2) x ((111n)!^(n-1)

          n times       n-1 times         n-2 times

First few values are...

1$ = 1

2$ = 22 x 2! = 22 x 2 = 44

3$ = 333 x 33! x 3!^2 = 2.081 Tredecillion

Now we get into the next type of factorial, which is megafactorial.

This was made by HaydenTheGoogologist2009.

Different symbol this time: n‽

_ = subscript

n‽ = a_n x a_n-1 x a_n-2 x ... x a_3 x a_2 x a_1

Here's some of the values:

1‽ = 1

2‽ = 4

3‽ = 18

4‽ = 4718592

5‽ = ~2.928 x 10^183237

Now, that may be mostly all of the factorials that currently exist, HOWEVER, I wouldn't be making this page if there was not any of MY versions of factorials, so lets go over them!

My version of megafactorial:

Symbol of choice: n@

Taking inspiration from Sloane and Plouffe's superfactorial and making another layer of that.

n@ = n$ x (n - 1)$ x (n - 2)$ x (n - 3)$ x ... x 3$ x 2$ x 1$

So, this means there's going to be a lot of calculating required...

Here's a few values I have calculated with this notation:

1@ = 1

2@ = 2

3@ = 24

4@ = 6912

5@ = 238878720

This is basically a higher tier version of Sloane and Plouffe's definition of superfactorial.

Now, you would think I would stop there, but NO.

Gigafactorial: n#

n# = n@ x (n - 1)@ x (n - 2)@ x (n - 3)@ x ... x 3@ x 2@ x 1@

This is the higher tier of my definition of megafactorial.

First few values:

1# = 1

2# = 2

3# = 48

4# = 331776

5# = 79254226206720

Terafactorial: n% = n# x (n - 1)# x (n - 2)# x (n - 3)# x ... x 3# x 2# x 1#

Petafactorial: n? = n% x (n - 1)% x (n - 2)% x (n - 3)% x ... x 3% x 2% x 1%

Etc, etc...

You may be seeing a pattern with this, HOW DO I KNOW WHAT TIER I'M IN??

This is where the next notation comes into play.

n}^x

(n = first number, x = second number)

So for example...

n}^1 is n!

n}^2 is n$

n}^3 is n@

n}^4 is n#

n}^5 is n?

etc.

More expansions...

n}^n}^2 = n}^^2}^2

n}^n}^n}^2 = n}^^3}^2

n}^^n}^2 = n}^^^2}^2

n}^^^^^2}^2 = n}(5)2}^2

n}}4}^2 = n}(n}(n}(n}(4)2})2})2})2}^2

Now, this is still not comparable to the first definition of superfactorial, but there is a fix to that.

Replace the curly bracket(s) with square bracket(s).

Curly brackets are for my defined tiers of what I like to call universalfactorial

Square brackets are for Pickover's defined tiers of universalfactorial

So by just changing the bracket type, we can create even larger numbers that are unimaginably large in size, in fact this could even compete with Graham's number!

n]^10 > n}^10

Let's just look at the tiers with the symbols real quickly...

n$ = n! ↑↑ n!

n@ = n$ ↑↑ n$

n# = n@ ↑↑ n@

n? = n# ↑↑ n#

More expansions using square brackets...

n]]n]]10]^2

n,2]10\4]^2 = n]]]]]]]]]]4]^2

Hey there, future me here, I was messing around with Wolfram Alpha when I may have potentially accidentally created a new factorial type that makes huge numbers.

It's Product[Product[n!,{n,1,n!}],{n,1,x}] with x being any number.

The symbol I'm going to represent it is ⍒.

0⍒= 1

1⍒= 1

2⍒= 2

3⍒= 49,766,400

4⍒= 5.84e250

5⍒= 9.17e10,776

6⍒= 2.00e584,577

In just only a few entries we overflow the standard computation time with this. Now sure how useful this is but it's fun for me to find new ways to create large numbers in only a few entries.

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