Exponential factorial

The exponential factorialExponential Factorial from Wolfram MathWorld or expofactorialA049384 from OEIS is an exponential version of the factorial, recursively defined as \(a_0 = 1\) and \(a_n = n^{a_{n - 1}}\). For example, \(a_6 = 6^{5^{4^{3^{2^1}}}}\).
The first few \(a_n\) for \(n = 0, 1, 2, 3, \ldots\) are 1, 1, 2, 9, 262,144, ... ().
The next entry, 5262,144, has 183,231 digits and starts with 6,206,069,878,660,874,470,748,320,557,284,67... Exponential factorial of 6 is approximately \(10^{4.829261036 \cdot 10^{183,230}}\) and starts with 1,103,560,225,917,696,632,179,145,334,475,34...
\(a_7\) is approximately \(10^{10^{4.82961036 \cdot 10^{183,230}}}\).
== Properties ==
The sum of the reciprocals of the exponential factorial numbers begins 2.6111149258083767361111...(183,213 1's)...1111272243... The long string of 1's appear because 1/9 = 0.111111111... and 1/2 and 1/262,144 have finite decimal expansions, and the reciprocal of is so small that more than 100,000 of the first decimal digits are zeroes.
The exponential factorial satisfies the bound \(a_n \leq {^{n-1}n}\) (tetration), and satisfies \(a_n < {^{n-k}n}\), for sufficiently large n (and any k).
It has a growth rate of approximately \(f_3(n)\) in the FGH.
In hyperfactorial array notation, the expofactorial of n can be written as n!1.
== Sources ==
== See also ==
*Expofaxul
*Zootzootplex
*Expostfacto function
*factorial
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