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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]

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

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.

• 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 $$g_{\varphi(\omega,0)}(n)$$

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