Facthype notation

Facthype notation (FHN) [NOT3]

• AKA Facthype function, Factorial Hyperoperation Notation

Updates

Update 4:

Moved from blog post and changed most of the text, the definition has not been altered

Update 5:

Extended page created, but there is nothing there yet

Update 6:

Updated to make stuff well-defined

This is a notation that i started developing when one day, i thought something along the lines of "what if the hyperoperators were based on the factorial?".

Most of the time, i divide my notation in "basics", "rules"/"features" and "extra features", so lets do that

NOTE: EVERY VARIABLE IS A NATURAL NUMBER UNLESS SPECIFIED OTHERWISE.

Basics:

Probably everyone reading this is familiar with nested factorial, but it is defined as:

n!^0 = n

n!^m = (n!^(m-1))!

Then we can build the function the notation is named after. This is the Factorial Hyperoperator Function:

n!^(1,m) = n!^m

n!^(k,m) = n!^(k-1,n!^(k,m-1))

n!^(k,0) = n

Yes, these are hypeoperators based on the factorial.

Of course, n!^(n,n) has a growth rate of ω in the fast growing hierarchy. We are now ready to move to the actual rules.

Rules:

First i will introduce some vocabulary:

The first argument (n in n!^(k,m)) is the "factoreo",

The second argument (k in n!^(k,m)) is the "great hyperfactor,"

The third argument (m in n!^(k,m)) is the "lesser hyperfactor."

The second and third argument can be referred as to "the hyperfactors" in conjunction

To clarify:

n(h)#(k,m)(a_1, a_2, . . . , a_(i-1), a_i) = n\#^{k,m}_{a_1,a_2,a_2,\cdots,a_{i-2},a_{i-1},a_i} (Latex code, Output)

This is because i can't use LaTeX on google sites (or at least i don't know how to) and you can't put a subscript and a hyperscript on the same place (at least i think), so yea.

1. n(1)#(k,m) = n!^(k,m)

2. n(h)#(a_1,a_2, . . . , a_(i-1), a_i, 1) = n(h)#(a_1,a_2, . . . , a_(i-1), a_i)

3. n(h)#(a_1,a_2, . . . , a_(i-1), a_i) =

   1. Let A_s = n + \sum_{k=1}^{i-s} a_{s-k+1} (Latex code, Output)

   2. n(h)#(a_1,a_2, . . . , a_(i-1), a_i) = n(h)#(A_1,A_2, . . . , A_(i-1), a_i - 1)

4. n(h)#(n,n)(n,n, . . (h-1 number of n's without counting the hyperfactors) . . ,n,n) = n#(h+1)(1)

5. n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i) =

   1. n(h)#(1)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(a_1, a_2, . . . , a_(i-1), a_i)

   2. n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i) = (n(h)#(m-1)(a_1, a_2, . . . , a_(i-1), a_i))(h)#(a_1, a_2, . . . , a_(i-1), a_i)

6. n(h)#(k,m)(a_1, a_2, . . . , a_(i-1), a_i)

   1. n(h)#(1,m)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i)

   2. n(h)#(k,m)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(k-1,n(h)#(k,m-1)(a_1, a_2, . . . , a_(i-1), a_i))(a_1, a_2, . . . , a_(i-1), a_i)

   3. n(h)#(k,1)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(a_1, a_2, . . . , a_(i-1), a_i)

7. n(h)#(k,m)(a + 1) = n(h)#(k-1,m)(n,n,n, . . (a times) . . ,n,n,n)

You may look at this and think to yourself "i didn't understand a sh*t" and it would be justified, this is just a bunch of rules mashed together, there isnt really an explanation behind it, there isnt even an order on when you should execute what. Although if you understood you can skip through the rules section.

Explanation

But first, more vocabulary:

All arguments but the factoreo, great and lesser hyperfactor are called operators.

Well lets continue

1. n(1)#(k,m) = n!^(k,m)

I don't think this one's hard to understand, but its just saying that n(1)#(k,m) is equivalent to n!^(k,m)

2. n(h)#(a_1,a_2, . . . , a_(i-1), a_i, 1) = n(h)#(a_1,a_2, . . . , a_(i-1), a_i)

Basically, if the last argument on the operators is 1, remove it.

3. n(h)#(a_1,a_2, . . . , a_(i-1), a_i) =

   1. Let A_s = n + \sum_{k=1}^{i-s} a_{s-k+1} (Latex code, Output)

   2. n(h)#(a_1,a_2, . . . , a_(i-1), a_i) = n(h)#(k,m)(A_1,A_2, . . . , A_(i-1), a_i - 1)

Let a be a sequence of natural numbers, with a_n being the nth number on that sequence and let i be the lenght of said sequence. Let A

be a sequence of natural numbers with A_s being the sth number in it = n + \sum_{k=1}^{i-s} a_{s-k+1} (Latex code, Output)

n(h)#(a_1,a_2, . . . , a_(i-1), a_i) = n(h)#(k,m)(A_1,A_2, . . . , A_(i-1), a_i - 1)

4. n(h)#(n,n)(n,n, . . (h-1 number of n's in the operators) . . ,n,n) = n#(h+1)(1)

Easy to understand

5. n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i)

   1. n(h)#(1)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(a_1, a_2, . . . , a_(i-1), a_i)

   2. n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i) = (n(h)#(m-1)(a_1, a_2, . . . , a_(i-1), a_i))(h)#(a_1, a_2, . . . , a_(i-1), a_i)

Let A be a sequence of natural numbers

n(h)#(m)(A) = (n(h)#(m-1)(A))(h)#(m-1)(A)

n(h)#(1)(A) = n(h)#(A)

Very much easier to read

6. n(h)#(k,m)(a_1, a_2, . . . , a_(i-1), a_i)

   1. n(h)#(1,m)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(m)(a_1, a_2, . . . , a_(i-1), a_i)

   2. n(h)#(k,m)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(k-1,n(h)#(k,m-1)(a_1, a_2, . . . , a_(i-1), a_i))(a_1, a_2, . . . , a_(i-1), a_i)

   3. n(h)#(k,1)(a_1, a_2, . . . , a_(i-1), a_i) = n(h)#(a_1, a_2, . . . , a_(i-1), a_i)

Let A be a sequence of natural numbers

n(h)#(k,m)(A) = n#(k-1,n#(k,m-1)(A))(A)

n(h)#(1,m)(A) = n(h)#(m)(A)

n(h)#(k,1) = n(h)#(A)

Also very much easier to read, but hey! This isn't over.

More stuff

Another thing: n! // m = n(m+1)#(1)

Although it hasn't been proven, i suspect n! // 3 has a growth rate of ω^ω! n! // 4 is intended to have a growth rate of ε_0, and this is also yet to be proved. The reason why i put this on "More stuff" instead of "Extra features" is because this will be important when I extend this notation.

Extra features

• n // ! is equal to n! // n, i dont know how fast it grows but it probably grows fast as hell, i will name it "superfactorial" to assert dominance in pickover and sloane. 2Slow4me, as some corporation trying to relate to a younger audience would say.

• I coin the name "destruction" and the notation n$ for the function n! // 2, which has a growth rate ω, n$ is pronounced as "n destroyed"

• I coin the name "Obliteration" and the notation n% (NOT TO BE CONFUSED WITH PERCENT) for the function n! // 3, which should have growth rate ω^ω. n% is pronounced as "n obliterated"

• I coin the name "Vanishment" and the notation n; for the function n! // 4, which should have growth rate ε_0. n; is pronounced as "n vanished"

• I coin the name "Interrogation" and the notation n? for the function n! // 5. n; is pronounced as "n sacrificed" (alternate: "n interrogated")

• Any function on the form of n! // m where m is some arbitrary natural can be referred alternatively as m-kaktorial

  • Ex. n! // 2 = "n bikaktorial", n! // 3 = "n trikaktorial", n! // 4 = "n quadkaktorial", n! // 5 "n quinkaktorial" and so on...

Special thanks to Binary198 for giving me the ideas to generalize up to n // !, and without him i wouldnt had made it so far