# 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
$$En\#\#n$$
$$\lbrace n,2,1,2 \rbrace$$ (exact value)
Fast-growing hierarchy $$f_\omega(n)$$
$$g_{\varphi(\omega,0)}(n)$$

## Sources Edit

1. Ackermann Number