Exponential factorial

An exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on and so forth, that is,

\( n^{(n - 1)^{(n - 2) \cdots }}.\, \)

The exponential factorial can also be defined with the recurrence relation

\( a_0 = 1,\quad a_n = n^{a_{n - 1}}.\,\)

The first few exponential factorials are 1, 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 5²⁶²¹⁴⁴ which is approximately 6.206069878660874 × 10¹⁸³²³⁰.

The sum of the reciprocals of the exponential factorials from 1 onwards is the transcendental number 1.6111149258083767361111... OEIS A080219.

Like tetration, there is currently no accepted method of extension of the exponential factorial function to real and complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function.

References

Jonathan Sondow, "Exponential Factorial" From Mathworld, a Wolfram Web resource

Mathematics Encyclopedia