Here, $\sigma(n)= \text{the sum of divisors of }n$.
For fun, I have been messing around on Mathematica with a particular arithmetical function, whose asymptotic behaviour I am interested in.
First, define $g(n)=\sigma(\sigma(n)) - \sigma(n)$. Next, define
$$s(n) = |\{ k:g(k)=n\}|.$$
Finally, our function:
$$S(n) = \sum_{1<i \leq n}s(i).$$
Now, some details are in order.
- $s(1)$ is infinite; ie. $g(k)=1$ whenever $\sigma(k)$ is prime.
- For $n>1$, $s(n)$ is finite. In fact, it is often zero.
I am interested in understanding the growth of $S(n)$. For small $n$, it appears linear. Below is a graph for $n\leq1600$.
It is definitely not linear though. Since we are working with $\sigma$, we can expect any closed-form expressions for, say, asymptotics, to have some double $\log$ terms.
Unfortunately I am a busy undergraduate student who knows nothing about analytic number theory! I don't have much work of my own, but I thought this was interesting enough to share. Is there a systematic approach to finding asymptotics for functions defined like $s,S$?
If we instead defined $s(n)$ to be the number of solutions to $\sigma(k)=n$, the corresponding $S(n)$ grows at a faster (and smoother) rate.