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 at 12:49

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