# Exponential factorial

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The exponential factorial[1] or expofactorial[2] 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, 262144, ... (OEIS A049384). The next number, 5262144, has 183231 digits and starts with 620606987866087447074832055728467... Exponential factorial of 6 is approximately $$10^{4.829261036 \cdot 10^{183230}}$$ and starts with 110356022591769663217914533447534....

The sum of the reciprocals of these numbers is 2.6111149258083767361111...(183,213 1's)...1111272243... The long string of 1's appear because 1/9 = 0.111111111... and 1/2 and 1/262144 have finite decimal expansions, and the reciprocal of 5262144 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).

In Hyperfactorial array notation, expofactorial is equal to n!1.