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 \(3\)\(\&\)\(n\) 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 20 digits

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

  • 1st = 1
  • 2nd = 4
  • 3rd = ...04,575,627,262,464,195,387 (tritri)
  • 4th = ...22,302,555,290,411,728,896 (tritet)
  • 5th = ...17,493,152,618,408,203,125 (tripent)
  • 6th = ...67,965,593,227,447,238,656 (trihex)
  • 7th = ...43,331,265,511,565,172,343 (trisept)
  • 8th = ...21,577,035,416,895,225,856 (trioct)
  • 9th = ...10,748,087,597,392,745,289 (triennet)
  • 10th = ...00,000,000,000,000,000,000 (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 \(f_\omega(n)\)
Slow-growing hierarchy \(g_{\varphi(\omega,0)}(n)\)


See also

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