Ackermann number

The Ackermann numbers are a sequence defined with the original definition of Ackermann function (not to be confused with the well-known Robinson's definition) as A(n) = A(n+2,n,n) where \(n\) is a positive integer. It can be expressed with arrow notation as:Conway and Guy. The Book of Numbers. Copernicus. 1995. ISBN 978-0387979939 p.60Ackermann Number
\[A(n) = n\underbrace{\uparrow\uparrow...\uparrow\uparrow}_nn\]
The first few Ackermann numbers are \(1\uparrow 1 = 1\), \(2\uparrow\uparrow 2 = 4\), and \(3\uparrow\uparrow\uparrow 3 =\) tritri. More generally, the Ackermann numbers diagonalize over arrow notation, and signify its growth rate is approximately \(f_\omega(n)\) in FGH and \(g_{\varphi(n-1,0)}(n)\) in SGH.
The \(n\)th Ackermann number could also be written \(3\)\(n\) or \(\lbrace n,n,n \rbrace\) in BEAF.
== Last 20 digits ==
Below are the last few digits of the first ten Ackermann numbers.Fish. Last 20 digits of Ackermann numbers 2024-08-29.
*1st = 1
*2nd = 4
*3rd = ...04575627262464195387 (tritri)
*4th = ...22302555290411728896 (tritet)
*5th = ...17493152618408203125 (tripent)
*6th = ...67965593227447238656 (trihex)
*7th = ...43331265511565172343 (trisept)
*8th = ...21577035416895225856 (trioct)
*9th = ...10748087597392745289 (triennet)
*10th = ...00000000000000000000 (tridecal)
== Approximations in other notations ==
Notation 	Approximation
Hyper-E notation 	\(\textrm En\#\#n\)
BEAF 	\(\lbrace n,2,1,2 \rbrace\) (exact value)
Fast-growing hierarchy	\(f_\omega(n)\)
Hardy hierarchy 	\(H_{\omega^\omega}(n)\)
Slow-growing hierarchy	\(g_{\varphi(\omega,0)}(n)\)
== Sources ==
== See also ==
*Arrow notation
*Ackermann function
*Ackermann's Generalized Exponential Notation
*Gag
ja:アッカーマン数
zh:阿克曼數
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