# 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

It doesn't have a commonly used name, no. You can call it a power tower of order 2 if you wish.

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Thanks! I was considering the context of complexity of algorithms. –  Tim Sep 24 '12 at 23:11
Hmm, "order" is used in many contexts, and maybe for that reason in the "tetration forum" we got used to say "height" instead of "order". That noun "height" can then be used in the more general context of iterating/selfcomposition of functions - it is also more colorful in context with "power tower" and so possibly is a good choice. (Then the letter "h" as an abbreviation comes also to mind) –  Gottfried Helms Sep 25 '12 at 7:45