Clarkkkkson

Clarkkkkson, denoted ¥, is a dynamic googologism that grows over time, based on the lynz.Further Lynz Explanation and the ClarkkkksonNotable Properties of Specific Numbers (page 22)
Let \(n!\) represent the factorial, and create the following extension:
* \(n!! = n! \cdot (n - 1)! \cdot (n - 2)! \cdot (n - 3)! \cdot \ldots\)
* \(n!!! = n!! \cdot (n - 1)!! \cdot (n - 2)!! \cdot (n - 3)!! \cdot \ldots\)
* \(n!!!! = n!!! \cdot (n - 1)!!! \cdot (n - 2)!!! \cdot (n - 3)!!! \cdot \ldots\)
* etc.
For the sake of this article, \(n!!\) will be abbreviated as \(n!2\), \(n!!!\) as \(n!3\), and so forth. The notation is due to Aalbert Torsius.
Define the hypf(c,p,n) function as follows (using down-arrow notation):
* \(\text{hypf}(c, 2, n) = n!c \cdot (n - 1)!c \cdot (n - 2)!c \cdot \ldots\)
* \(\text{hypf}(c, 3, n) = n!c \downarrow (n - 1)!c \downarrow (n - 2)!c \downarrow \ldots\)
* \(\text{hypf}(c, 4, n) = n!c \downarrow\downarrow (n - 1)!c \downarrow\downarrow (n - 2)!c \downarrow\downarrow \ldots\)
* and so on through the weak hyper-operators.
Further, define the Clarkkkkson function ck(c, p, n, r) = hypf(c, p, ck(c,p,n,r-1)) and ck(c,p,n,1)=hypf(c,p,n).
Finally, define the following series:
* \(A_{1} = ck(K, K, K, K)\) (where \(K\) is the lynz)
* \(A_{2} = ck(A_{1}, A_{1}, A_{1}, A_{1})\)
* \(A_{3} = ck(A_{2}, A_{2}, A_{2}, A_{2})\)
* etc.
* The Clarkkkkson is AK + 1.
The Clarkkkkson changes over time. Thus, it can be considered a function of time instead of a true number.
The Clarkkkkson passed Graham's number shortly after it was defined.
Its current value ( for UTC) ([ refresh]) is approximately fω+1(), using the fast-growing hierarchy.
== Approximations ==
This number grows over time, and so does the approximation.
Notation 	Approximation
Hyper-E notation 	E10##10##
Chained arrow notation 	10→10→→2
BEAF 	{10,,1,2}
Strong array notation 	s(10,,2,2)
Fast-growing hierarchy (using CNF's fundamental sequences)	fω+1()
Hardy hierarchy 	Hωω+1()
Slow-growing hierarchy 	gΓ0()
== Sources ==
== See also ==
* Lynz
* Clarkkkksonplex
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Category:Dynamic googolisms