When I graph the sum of each number in the prime factorization of n, I get a strange graph. The individual values seem random, but it definitely has a pattern. Do we know why this is, and if so, why?
To be clear, I'm summing like this:
$f(36) = (2 + 2 + 3 + 3) = 10$
rather than
$f(36) = (2^2 + 3^2) = 13$
furthermore, not only does it have an overall linear slope, it appears to have (at least) 3 smaller lines, highlighted here:
I'm guessing that we don't know exactly why, but even then, might there be some vague hints that we've discovered, at least?
Notice that for $p\in\mathbb{P}$, $f(p)=p$, so the primes map linearly with slope $1$.
Then let $k\in\mathbb{Z}^+$ and consider: $$f(kp)=f(k)+f(p)=f(k)+p$$ The slope isn't readily obvious for this one, though, so notice that $$k\cdot(p+m)=kp+km$$ and that $$f(k\cdot(p+m))=f(k)+p+m$$ whenever $p+m\in\mathbb{P}$. So the slope will be $\frac{m}{km}=\frac1k$ between $kp$ and $k\cdot(p+m)$. Since the slope is independent of $p$ or $m$, you have the slope for the line related to $k$.
The pattern arises simply because the function sort of "plays nicely" with the primes.