This question concerns the behavior of a function $f(\;)$ that maps each number in $\mathbb{N}$ to its mean prime factor. I previously posted premature questions, now deleted, which explains the cites below to several who contributed observations. Also, the question, "Distribution of the number of prime factorsDistribution of the number of prime factors," may be relevant.
Define $f(n)$ to be the floor of the geometric mean of all the prime factors of $n$. For example, $$n = 6500 = 2^2 \cdot 5^3 \cdot 13 = 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 13 \;.$$ $$f(6500) = \lfloor\left( 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 13 \right)^{\frac{1}{6}}\rfloor = \lfloor 4.32 \rfloor = 4 \;. $$ For $n$ a prime, $f(n) = n$, i.e., the primes are fixed points of $f(\;)$. For $n$ a prime power $p^k$, $f(n)=p$ (as Julian Rosen observed). Greg Martin noted that $$ f(n) = \lfloor n ^ {1/\Omega(n)} \rfloor $$ where $\Omega(n)$ is the number of primes dividing $n$ counted with multiplicity.
I explored $|f^{-1}(k)|$, the number of $n \in \mathbb{N}$ that map to $k$: $f(n)=k$. The root transition points are crucial (as Gerhard Paseman emphasized). For $n_\max=10^7$, the square-root, cube-root, and fourth-roots of $n_\max$ are $(3163,216,57)$ rounded up. Here are graphs of $|f^{-1}|$ at two scales:
[![Sqrt][1]][1]
$n_\max=10^7$. Note the transitions at $216$ and $3163$.
[![Cbrt][2]][2]
$n_\max=10^7$. Note the transitions at $57$ and at $216$.
Q. What explains the apparent linearity in the first graph in the range $[216,3163]$, in contrast to the gradual upsweep in the range $[57,216]$ in the second graph?
I did not expect to see such regularity...
