Prime factorization "demoted" leads to function whose fixed points are primes

A way to get a non-trivial solution to $f(n) = p$ is that every odd number $\geq 7$ can be written as a sum of three primes (by Helfgott's recent work), so if $p \geq 7$ is prime, we can write $p = q + r + s$, and we have $g(qrs) = q + r + s = p$. (This is of course a bit overkill, we don't really need such a difficult result to see this.)

NOT AN ANSWER, just an illustration :D ($g$ up to $n=150$)
Quite amusing...

In case anybody wants to play with this, here is the Mathematica code

g[n_] := Dot @@ Transpose[FactorInteger[n]]
Graph[Map[# \[DirectedEdge] g[#] &, Range[2, 150]],
    VertexLabels -> Placed["Name", {1/2, 1/2}], VertexShape -> ""]

And since, as discussed in the comments below, the picture might be misleading in that cutting at any given $n$ creates false impression that larger numbers have smaller $g$-preimages while in fact it is the exact opposite, here is the table of sizes and smallest and largest elements in $g^{-1}(n)$ for $n\leqslant30$:

n   |  min size max
-------------------
2   |   2   1   2
3   |   3   1   3
4   |   4   1   4
5   |   5   2   6
6   |   8   2   9
7   |   7   3   12
8   |   15  3   18
9   |   14  4   27
10  |   21  5   36
11  |   11  6   54
12  |   35  7   81
13  |   13  9   108
14  |   33  10  162
15  |   26  12  243
16  |   39  14  324
17  |   17  17  486
18  |   65  19  729
19  |   19  23  972
20  |   51  26  1458
21  |   38  30  2187
22  |   57  35  2916
23  |   23  40  4374
24  |   95  46  6561
25  |   46  52  8748
26  |   69  60  13122
27  |   92  67  19683
28  |   115 77  26244
29  |   29  87  39366
30  |   161 98  59049

And here is the code for $g^{-1}$:

ginverse[n_]:=Which[
    n == 0, {1},
    n == 1, {},
    n == 2, {2},
    True, With[{p = Prime[Range[PrimePi[n]]]}, 
        Sort[Map[Times @@ (p^#) &, FrobeniusSolve[p, n]]]]]

Here is a (portion of a) histogram of $|f^{-1}(n)|$ for $n=5,\ldots,10^6$ (newly updated from $10^5$ to $10^6$):

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$266429$ of those numbers $n \le 10^6$ map $f(n)=5$; $152548$ map $f(n)=7$.