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Not to be confused with Ackermann ordinal.

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:[1][2]

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

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.[3]

• 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

1. Conway and Guy. The Book of Numbers. Copernicus. 1995. ISBN 978-0387979939 p.60
2. Ackermann Number
3. Fish. Last 20 digits of Ackermann numbers 2024-08-29.