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NOTE: THIS BLOG POST IS OBSOLETE, UPDATES WILL BE MADE ON HERE
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Facthype Notation [ ] Facthype notation (Short for Factorial Hyperoperator Notation, of which the acronym is F.H.N.
Facthype notation
Alternate names
Facthype function, Factorial hyperoperators
Notations
\(n\#^{k,m}_{a_1,a_2,a_2,\cdots,a_{i-2},a_{i-1},a_i}\)
Updates [ ] Update 1 (20/01/22) [ ] Added coined googolisms and functions sections.
Added (currently empty) extra features section.
Added Updates section.
Extended the definition. Update 1.1 (25/01/22) [ ] Small corrections. Update 2 (Also 25/01/22) [ ] Expanded defintion. (Credits to User:Binary198 )
Added some stuff to the extra features section. Update 2.1 (Also 25/01/22) [ ] Added more stuff to the googolisms section. Update 2.2 (Also 25/01/22) [ ] It is ill-defined (pain) but i tried fixing some errors and i think everything before # is well-defined now (Edit: its not), although i don't if the growth rates are correct though. Update 3 (26/01/22) [ ] I dont know if it is still ill-defined, but i fixed some stuff.
Expanded definition, and slight change to %.
Added "Unknown" section.
Moved function from "Extra features" to "Coined functions".
Added new function to "Coined functions". Update 3.1 (Also 26/01/22) [ ] Added infobox.
Fixed typo. Update 3.1.1 (Also 26/01/22) [ ] Very small changes on the # section. Update 3.1.2 (Also 26/01/22) [ ] Added more to the infobox. Update 4 [ ] Blog post obsolete, moving here . Definition [ ] Basic rules [ ] Note :
Let
n
!
m
=
{
n
!
,
for
m
=
1
(
n
!
m
−
1
)
!
,
for
m
>
1
{\displaystyle n!^{m}={\begin{cases}n!,&{\text{for }}m=1\\(n!^{m-1})!,&{\text{for }}m>1\end{cases}}}
Now let
n
!
k
,
m
=
{
n
!
m
,
for
k
=
1
n
!
k
−
1
,
n
!
k
−
1
,
m
−
1
,
for
k
>
1
and
m
>
1
n
!
,
for
m
=
1
{\displaystyle n!^{k,m}={\begin{cases}n!^{m},&{\text{for }}k=1\\n!^{k-1,n!^{k-1,m-1}},&{\text{for }}k>1{\text{ and }}m>1\\n!,&{\text{for }}m=1\end{cases}}}
n
!
n
,
n
≈
f
ω
(
n
)
{\displaystyle n!^{n,n}\approx f_{\omega }(n)}
Features [ ] $ [ ]
n
$
=
n
!
n
,
n
{\displaystyle n\$=n!^{n,n}}
n
$
m
=
{
n
$
,
for
m
=
1
(
n
$
m
−
1
)
$
,
for
m
>
1
{\displaystyle n\$^{m}={\begin{cases}n\$,&{\text{for }}m=1\\(n\$^{m-1})\$,&{\text{for }}m>1\end{cases}}}
n
$
k
,
m
=
{
n
$
m
,
for
k
=
1
n
$
k
−
1
,
n
$
k
−
1
,
m
−
1
,
for
k
>
1
and
m
>
1
n
$
,
for
m
=
1
{\displaystyle n\$^{k,m}={\begin{cases}n\$^{m},&{\text{for }}k=1\\n\$^{k-1,n\$^{k-1,m-1}},&{\text{for }}k>1{\text{ and }}m>1\\n\$,&{\text{for }}m=1\end{cases}}}
n
$
n
,
n
≈
f
ω
2
(
n
)
{\displaystyle n\$^{n,n}\approx f_{\omega 2}(n)}
n
$
1
=
n
$
{\displaystyle n\$_{1}=n\$}
n
$
h
m
=
{
n
$
h
,
for
m
=
1
(
n
$
h
m
−
1
)
$
k
,
for
m
>
1
{\displaystyle n\$_{h}^{m}={\begin{cases}n\$_{h},&{\text{for }}m=1\\(n\$_{h}^{m-1})\$_{k},&{\text{for }}m>1\end{cases}}}
n
$
h
k
,
m
=
{
n
$
h
m
,
for
k
=
1
n
$
h
k
−
1
,
n
$
h
k
−
1
,
m
−
1
,
for
k
>
1
{\displaystyle n\$_{h}^{k,m}={\begin{cases}n\$_{h}^{m},&{\text{for }}k=1\\n\$_{h}^{k-1,n\$_{h}^{k-1,m-1}},&{\text{for }}k>1\end{cases}}}
n
$
k
=
n
$
k
−
1
n
,
n
{\displaystyle n\$_{k}=n\$_{k-1}^{n,n}}
n
$
n
n
,
n
≈
f
ω
2
(
n
)
{\displaystyle n\$_{n}^{n,n}\approx f_{\omega ^{2}}(n)}
% [ ]
n
%
=
n
$
n
n
,
n
{\displaystyle n\%=n\$_{n}^{n,n}}
n
%
1
=
n
%
{\displaystyle n\%_{1}=n\%}
n
%
1
,
1
=
n
%
1
{\displaystyle n\%_{1,1}=n\%_{1}}
n
%
h
,
i
m
=
{
n
%
h
,
i
,
for
m
=
1
(
n
%
h
,
i
m
−
1
)
%
h
,
i
,
for
m
>
1
{\displaystyle n\%_{h,i}^{m}={\begin{cases}n\%_{h,i},&{\text{for }}m=1\\(n\%_{h,i}^{m-1})\%_{h,i},&{\text{for }}m>1\end{cases}}}
n
%
h
,
i
k
,
m
=
f
(
n
)
=
{
n
%
h
,
i
m
,
for
k
=
1
n
%
h
,
i
k
−
1
,
n
%
h
,
i
k
−
1
,
m
−
1
,
for
k
>
1
and
m
>
1
n
%
h
,
i
,
for
m
=
1
{\displaystyle n\%_{h,i}^{k,m}=f(n)={\begin{cases}n\%_{h,i}^{m},&{\text{for }}k=1\\n\%_{h,i}^{k-1,n\%_{h,i}^{k-1,m-1}},&{\text{for }}k>1{\text{ and }}m>1\\n\%_{h,i},&{\text{for }}m=1\end{cases}}}
n
%
h
,
i
=
n
%
h
−
1
,
i
n
,
n
{\displaystyle n\%_{h,i}=n\%_{h-1,i}^{n,n}}
n
%
h
,
i
=
n
%
n
+
h
,
i
−
1
n
,
n
{\displaystyle n\%_{h,i}=n\%_{n+h,i-1}^{n,n}}
n
%
n
,
n
n
,
n
≈
f
ω
ω
(
n
)
{\displaystyle n\%_{n,n}^{n,n}\approx f_{\omega ^{\omega }}(n)}
# [ ]
n
(
1
)
#
=
n
!
{\displaystyle n(1)\#=n!}
n
(
1
)
#
m
=
n
!
m
{\displaystyle n(1)\#^{m}=n!^{m}}
n
(
1
)
#
k
,
m
=
n
!
k
,
m
{\displaystyle n(1)\#^{k,m}=n!^{k,m}}
n
(
h
)
#
a
1
,
a
2
,
⋯
,
a
i
−
1
,
a
i
⏟
i
,
1
k
,
m
=
n
(
h
)
#
a
1
,
a
2
,
⋯
,
a
i
−
1
,
a
i
⏟
i
k
,
m
{\displaystyle n(h)\#_{\underbrace {a_{1},a_{2},\cdots ,a_{i-1},a_{i}} _{i},1}^{k,m}=n(h)\#_{\underbrace {a_{1},a_{2},\cdots ,a_{i-1},a_{i}} _{i}}^{k,m}}
Let
A
=
(
a
1
,
a
2
,
⋯
,
a
i
−
1
,
a
i
)
{\displaystyle A=(a_{1},a_{2},\cdots ,a_{i-1},a_{i})}
n
(
h
)
#
A
m
=
{
n
(
h
)
#
,
for
m
=
1
(
n
(
h
)
#
A
m
−
1
)
(
h
)
#
A
,
for
m
>
1
{\displaystyle n(h)\#_{A}^{m}={\begin{cases}n(h)\#,&{\text{for }}m=1\\(n(h)\#_{A}^{m-1})(h)\#_{A},&{\text{for }}m>1\end{cases}}}
n
(
h
)
#
A
k
,
m
=
{
n
(
h
)
#
A
m
,
for
k
=
1
n
(
h
)
#
A
k
−
1
,
n
(
h
)
#
A
k
−
1
,
m
−
1
,
for
k
>
1
and
m
>
1
n
(
h
)
#
A
,
for
m
=
1
{\displaystyle n(h)\#_{A}^{k,m}={\begin{cases}n(h)\#_{A}^{m},&{\text{for }}k=1\\n(h)\#_{A}^{k-1,n(h)\#_{A}^{k-1,m-1}},&{\text{for }}k>1{\text{ and }}m>1\\n(h)\#_{A},&{\text{for }}m=1\end{cases}}}
n
(
h
)
#
a
1
,
a
2
,
⋯
,
a
i
−
1
,
a
i
⏟
i
k
,
m
=
{\displaystyle n(h)\#_{\underbrace {a_{1},a_{2},\cdots ,a_{i-1},a_{i}} _{i}}^{k,m}=}
Let
B
s
=
n
+
∑
k
=
1
i
−
s
a
s
−
k
+
1
{\displaystyle {\text{Let }}B_{s}=n+\sum _{k=1}^{i-s}a_{s-k+1}}
n
(
h
)
#
a
1
,
a
2
,
⋯
,
a
i
−
1
,
a
i
⏟
i
k
,
m
=
n
(
h
)
#
B
1
,
B
2
,
⋯
,
B
i
−
1
⏟
i
,
a
i
−
1
k
,
m
{\displaystyle n(h)\#_{\underbrace {a_{1},a_{2},\cdots ,a_{i-1},a_{i}} _{i}}^{k,m}=n(h)\#_{\underbrace {B_{1},B_{2},\cdots ,B_{i-1}} _{i},a_{i}-1}^{k,m}}
n
(
h
)
#
=
n
(
h
−
1
)
#
n
,
n
,
n
,
⋯
,
n
,
n
,
n
⏟
h
−
1
n
,
n
for
h
>
1
{\displaystyle n(h)\#=n(h-1)\#_{\underbrace {n,n,n,\cdots ,n,n,n} _{h-1}}^{n,n}{\text{ for }}h>1}
n
(
4
)
#
(
n
,
n
,
n
,
n
,
n
)
≈
f
ϵ
0
(
n
)
{\displaystyle n(4)\#(n,n,n,n,n)\approx f_{\epsilon _{0}}(n)}
n
!
[
m
]
=
n
(
m
)
#
n
,
n
,
⋯
,
n
,
n
⏟
m
n
,
n
{\displaystyle n![m]=n(m)\#_{\underbrace {n,n,\cdots ,n,n} _{m}}^{n,n}}
[ ] Googolisms [ ] Generated by prefixes and suffixes [ ] -Obliteration (n%) [ ] Triobliteration - 3%
Quadobliteration - 4%
Quinobliteration - 5%
Sexobliteration (unfortunate name) - 6%
Triobliterationobliteration - (3%)% -Hyteration (n%^n%) [ ] Trihyteration -
3
%
3
%
{\displaystyle 3\%^{3\%}}
Quadhyteration -
4
%
4
%
{\displaystyle 4\%^{4\%}}
Quinhyterion -
5
%
5
%
{\displaystyle 5\%^{5\%}}
-Erased (n;) [ ] Trierased -
3
;
{\displaystyle 3;}
Quaderased -
4
;
{\displaystyle 4;}
Quinerased -
5
;
{\displaystyle 5;}
Sexerased -
6
;
{\displaystyle 6;}
Named [ ] Grahamtorial -
3
$
64
{\displaystyle 3\$^{64}}
Factorial God- 170?
Godly 3 -
3
%
33
,
333
33
,
333
{\displaystyle 3\%_{33,333}^{33,333}}
Hierarchy -
50
?
50
%
50
$
50
!
{\displaystyle 50?^{50\%^{50\$^{50!}}}}
Hieranarchy -
50
(
4
)
#
50
%
,
50
?
,
50
;
50
!
,
50
$
{\displaystyle 50(4)\#_{50\%,50?,50;}^{50!,50\$}}
Misfact -
Let
f
(
n
)
=
3
!
[
n
]
{\displaystyle {\text{Let }}f(n)=3![n]}
Let
f
m
(
n
)
=
{
f
(
n
)
,
for
m
=
1
f
(
f
m
−
1
(
n
)
)
,
for
m
>
1
{\displaystyle {\text{Let }}f^{m}(n)={\begin{cases}f(n),&{\text{for }}m=1\\f(f^{m-1}(n)),&{\text{for }}m>1\end{cases}}}
Misfact
=
M
I
S
F
=
f
33
[
!
]
(
3
)
{\displaystyle {\text{Misfact}}=MISF=f^{33[!]}(3)}
Coined functions [ ] Fact-operating - n!^(k,m) - Said as: "n (k+3)-torial"
a!^(1,b) - a tetratorial to b
a!^(2,b) - a pentatorial to b
a!^(3,b) - a hexatorial to b
etc.
Destruction -
n
$
{\displaystyle n\$}
Interrogation -
n
?
=
n
%
n
,
n
n
,
n
{\displaystyle n?=n\%_{n,n}^{n,n}}
n
?
m
=
n
?
?
⋯
?
?
⏟
m
{\displaystyle n?^{m}=n\underbrace {??\cdots ??} _{m}}
Vanishment -
n
;
=
n
(
3
)
#
n
,
n
,
n
n
,
n
{\displaystyle n;=n(3)\#_{n,n,n}^{n,n}}
n
;
m
=
n
;
;
⋯
;
;
⏟
m
{\displaystyle n;^{m}=n\underbrace {;;\cdots ;;} _{m}}
Superfactorial -
n
[
!
]
=
n
!
[
n
]
{\displaystyle n[!]=n![n]}
n
[
!
]
m
=
n
[
!
]
[
!
]
[
!
]
⋯
[
!
]
[
!
]
[
!
]
⏟
m
{\displaystyle n[!]^{m}=n\underbrace {[!][!][!]\cdots [!][!][!]} _{m}}
Unknown [ ] Growth rates [ ] (NOTE THAT SINCE THE NOTATION IS ILL-DEFINED THESE ARE MEANINGLESS)
n$^{n,n} - Its most probably w2, but it remains unproven. n% - its most probably w^2, but it remains unproven. n? - Its most probably w^w, but it remains unproven. n; - No one knows really whats more probable, but it is intended to have a growth rate of e_0. n![m] for m>4 - Totally unknown.
n[!] - Totally unknown.
![{\displaystyle n![m]=n(m)\#_{\underbrace {n,n,\cdots ,n,n} _{m}}^{n,n}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/8c87a869224687841d826eb545a936007481117a)
![{\displaystyle {\text{Let }}f(n)=3![n]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/b0ba892651d1b583d279801ef2e3cdc4137e9a82)
![{\displaystyle {\text{Misfact}}=MISF=f^{33[!]}(3)}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/fa378fb45cba60097df933a9f0b0fd653a5a463e)
![{\displaystyle n[!]=n![n]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a04e6aa528382990eaed78ec376de98445f4116e)
![{\displaystyle n[!]^{m}=n\underbrace {[!][!][!]\cdots [!][!][!]} _{m}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/d7e07385170df4ac31b054b3b63b4cd6ba35cc8e)