If, for positive integer $n=p_1^{q_1}p_2^{q_2}\dots p_k^{q_k}$ with the right side the prime factorization of $n$, we define $J(n)$ as $p_1q_1+p_2q_2+\dots+p_kq_k$, does $J(n)$ have any generally "interesting" properties?
I first thought of this function while idly imagining pseudovariants of logarithms that would only make sense on the natural numbers. We can apply $J(n)$'s definition to verify that $J(ab)=J(a)+J(b)$ and note that this function is indeed, sort of, in a sense, logarithmic, just with a "base" that varies depending on what $n$ is.
For an example of a potentially interesting property, $\frac{J(n)}{n}$ (which I will hereafter call $j(n)$ to save space) is greatest when $n$ is prime and secondarily when $n$ is dominated by one large prime factor (for example, $j(n)=1$ when $n$ is prime and $j(n)\approx 1+3\cdot 10^{-5}$ when $n=2\times 65537$), and $j(n)$ is least when $n$ has lots of small factors (for example, $j(n)\approx 2\cdot 10^{-4}$ when $n=2^5 3^3 5^2 7^1$ and $j(n)\to 0$ when $n=2^k$ for integer $k$)—by checking small $n$ manually in Desmos (note: graph edited to take advantage of Qiaochu Yuan's answer, then to remove $n=1$ because I realized that that case is unpleasant), I found that $j(n)$ has minima for $n=6,8,9,12,15,16,20,25,27,32$. As such, $0<j(n)\le 1$, and these are the best possible constant bounds (though presumably a better lower bound depending on $n$ somehow could be determined). Further interesting properties could take pretty much any form you personally think appropriate, which I appreciate might make this problem too soft to be strictly appropriate, but I considered it an interesting function and would be interested to hear anything others come up with.
Also, could someone please graph this function for $n\le 1000$ or so? I tried to get Wolfram Alpha to do it, but didn't know how to phrase my request in a way that gave an answer where it wasn't just interpreting it as "$p_1q_1+p_2q_2$". I assume Mathematica or something like that (other programming languages would also presumably be usable) would have a better shot at getting something like the result I'm hoping for, but I don't know how to use Mathematica.
The functions $q_i$ are individually interesting and the subject of study; generally, if $p$ is prime then $\nu_p(n)$, the $p$-adic valuation of $n$, is the exponent of $p$ appearing in the prime factorization of $n$. It also has the logarithmic property $\nu_p(nm) = \nu_p(n) + \nu_p(m)$, and it's not hard to show that in fact any logarithmic function on the natural numbers (although apparently the standard term is actually "completely additive") is a linear combination (possibly infinite) of the $p$-adic valuations $\nu_p$.
Your function $J(n)$ is the specific linear combination $\sum p \nu_p(n)$; it is A001414 on the OEIS and you can check that link for more information. It is apparently sometimes called the "integer logarithm" or the "potency." The OEIS has a graph:
It also records the lower bound
$$J(n) \ge 3 \log_3 n$$
which is tight when $n = 3^k$ is a power of $3$.
The related linear combination $\sum \nu_p(n)$ records the total number of prime factors of $n$ with multiplicity and is A001222 on OEIS; this is the "big" prime omega function $\Omega(n)$ and you can check that link for various facts about it. I think it has probably been studied more extensively than $J$.