Ackermann Number

The Ackermann numbers are a sequence defined using arrow notation as:

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

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 nth 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.

Contents

• Last 20 digits
• Approximations in other notations
• Sources
• See also

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)

Sources

https://mathworld.wolfram.com/AckermannNumber.html

External Links

• https://googology.wikia.org/wiki/Ackermann_number
• https://googologytesting.fandom.com/wiki/Ackermann_number