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Ackermann number

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The Ackermann numbers are a sequence defined using Arrow Notation as[1]

\[A(n) = n\underbrace{\uparrow\uparrow...\uparrow\uparrow}_nn\]

where \(n\) is a positive integer. 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.

The Ackermann numbers are related to the Ackermann function; they exhibit similar growth rates, although their definitions are quite different.

Last 10 digits Edit

Below are the last few digits of the first ten Ackermann numbers.

Approximations in other notations Edit

Notation Approximation

Hyper-E notation



\(\lbrace n,2,1,2 \rbrace\) (exact value)
Fast-growing hierarchy \(f_\omega(n)\)

Slow-growing hierarchy


Sources Edit

  1. Ackermann Number

See also Edit

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