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:
$f(36) = (2 + 2 + 3 + 3) = 10$
$f(36) = (2^2 + 3^2) = 13$
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?