Louvre Pyramid

2014-01-24T14:58:32Z

146.201.213.39: Changed height from 20.6m/70 ft (an incorrect conversion) to 21.6m/71ft, which matches the referenced source and other external sources that do not cite Wikipedia. 20.6 was probably a typographical error.

An '''exponential factorial''' is a positive integer ''n'' [[exponentiation|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,

: <math>n^{(n - 1)^{(n - 2) \cdots }}.\,</math>

The exponential factorial can also be defined with the [[recurrence relation]]

: <math>a_0 = 1,\quad a_n = n^{a_{n - 1}}.\,</math>

The first few exponential factorials are [[1 (number)|1]], [[1 (number)|1]], [[2 (number)|2]], [[9 (number)|9]], 262144, etc. {{OEIS|id=A049384}}. So, for example, 262144 is an exponential factorial since

: <math>262144 = 4^{3^{2^{1}}}.\,</math>

The exponential factorials grow much more quickly than regular [[factorial]]s or even [[Factorial#Hyperfactorial|hyperfactorial]]s. The exponential factorial of 5 is 5<sup>262144</sup> which is approximately 6.206069878660874 × 10<sup>183230</sup>.

The sum of the reciprocals of the exponential factorials from 1 onwards is the [[transcendental number]] 1.6111149258083767361111... {{OEIS2C|id=A080219}}.

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

==References==
*Jonathan Sondow, "[http://mathworld.wolfram.com/ExponentialFactorial.html Exponential Factorial]" From [[Mathworld]], a Wolfram Web resource

[[Category:Factorial and binomial topics]]
[[Category:Integer sequences]]
[[Category:Large integers]]

{{Numtheory-stub}}

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