While working with numbers of the form $p_n\#$ (the primorials) I began to wonder about what an inverse of the "primorial" operation might be like. I think that reusing the symbol makes some sense, so we might say something like
$\#^{-1}(n)$ is the largest prime factor of $n$.
With this definition, we can get some simple properties:
- $(x,(\#^{-1}(x))\#)$ is the largest squarefree number which divides $x$
- $p_n=\#^{-1}(p_n\#)$ for every $n$
- $\forall z\in \Bbb Z^+,\#^{-1}(z\#)=p_{\pi(z)}$ is the largest prime not greater than $z$.
Seeing as $\pi(n)$ and $p_n$ already provide highly useful functions, it seems that this particular function has very limited usefulness. Are there other known uses out there for this function? Is there already a given name and/or symbol for this function?