Exponential factorial

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An exponential factorial is a positive integer n raised to the power of n - 1, which in turn was raised to the power of n - 2, and so on and so forth, that is, $n^{(n - 1)^{(n - 2) \dots }}$ . The exponential factorial can also be defined with the recurrence relation $a_0 = 1, a_n = n^{a_{n - 1}}$ .

The first few exponential factorials are 1, 2, 9, 262144, etc. (sequence A049384 in OEIS). So, for example, 262144 is an exponential factorial since $262144 = 4^{3^{2^1}}$ . The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The exponential factorial of 5 is approximately 6.206069878660874 × 10¹⁸³²³⁰.