Remember σ(pn) = p * σ(n) + σ(n / p^α)?

More generally, σ(p^kn) = p^k * σ(n) + (p^k – 1) / (p – 1) * σ(n / p^α).

Again, p is a prime and alpha is its multiplicity in n’s prime factorization. σ is the divisor function, of course.

All I need to do now is figure out how to extend it to a composite number and I’ll have a complete multiplicative recurrence on the divisor function, which I can use to obtain a closed-form rate of growth. I’ve empirically calculated it to grow at approximately 1.6449*n, but my goal is to obtain a tight worst-case bound. I could not find anything special about this number, except that is the 90% critical value of a normal distribution.

Here’s a messy Maple worksheet containing the derivation (among a whole bunch of stuff not related to the derivation that I was experimenting with today).