# Is there a name for $n^n$?

As functions of $n$, $n^c$ is called power, and $e^n$ is called exponential. Is there a name for $n^n$ as a function of $n$? Thanks!

Added: consider the context of complexity of algorithms.

Also is $n^n$ an elementary function?

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This is also ${}^2n$, where the notation indicates the fourth hyper operator, which is most often called "tetration." Annoyingly, I don't know of any good way to read a tetration out loud-something like "$n$ tetrated twice" or "$n$ tetrate two", I suppose-or even "$n$ tetrated" as the special case of $n^n$.

But as I say, you'd have to explain yourself on any of these, so it's unlikely to be of any use, except maybe in the midst of a talk where you don't want to repeat "$n$ to the $n$" a dozen times. The hyper operators are mainly useful for simplifying notation of extraordinarily large numbers, so in day-to-day complexity theory they might not be of much use.

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Thanks! (1) Are you saying $^2n$ means $n^n$? (2) Is $n^n$ an elementary function? –  Tim Sep 25 '12 at 0:24
@Tim (1) yes that is what he means (2) yes: $n^n=e^{n\ln n}$ –  user39572 Sep 25 '12 at 0:26
Hmm, I think this is a bit too much for the actual question. The term "tetration" is used in the case when someone considers the iterated exponentiation with respect to a variable "height" parameter. But as I understand the question here it is only asked for the specific case of the function $n^n$. I've seen sometimes "self power" for this (but I think it is an awful name...) –  Gottfried Helms Sep 25 '12 at 7:49