# Is There an Operation Between Exponentiation and Tetration?

The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as \begin{align*} \text{exp} (a, n) = a^n=a\cdot a\cdots a \end{align*} and tetration as \begin{align*} \text{tet}(a, n)&=\text{exp} (\text{exp}(\cdots \text{exp}(a, n)))\\ &=a^{a^{\cdots^{a} } } \end{align*} where $$a$$ and $$n$$ are integers, then is there a type of operation that falls between these two? In other words, is there a function $$f(n)$$ such that $$\text{exp} (a, n) < f(n) < \text{tet}(n, a)$$ for (most) integers $$n$$? My guess would be the factorial function but I'm not sure.

• Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $\operatorname{tet}(n,b)$ for fixed $b$. – Dark Malthorp Dec 3 '18 at 12:49