*Not to be confused with Ackermann ordinal.*

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 \(3\)Template:Mathlink\(n\) 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.

- 1st = 1
- 2nd = 4
- 3rd = ...2,464,195,387 (tritri)
- 4th = ...0,411,728,896 (tritet)
- 5th = ...8,408,203,125 (tripent)
- 6th = ...7,447,238,656 (trihex)
- 7th = ...1,565,172,343 (trisept)
- 8th = ...6,895,225,856 (trioct)
- 9th = ...7,392,745,289 (trienn)
- 10th = ...0,000,000,000 (tridecal)