Behavior of the “mean prime factor” of numbers

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 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:

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^{$n_\max=10^7$. Note the transitions at $216$ and $3163$.}

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^{$n_\max=10^7$. Note the transitions at $57$ and at $216$.}

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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…

Answer

Attribution
Source : Link , Question Author : Joseph O’Rourke , Answer Author : Community