What is known about iterative behavior of this function? Let $p$ be a function that sums divisors of some natural number but which does not sum that number.
For example we have $p(12)=1+2+3+4+6=16$.
We can take that $p(1)=1$ to avoid possible difficulties later.
Perfect numbers are fixed points of this function, and for primes we have $p(q)=1$ and $p(p(q)))=1$ if $q$ is a prime, so, on perfects and primes, this function has a very simple behavior.
It would be nice to know how this function behaves with respect to iteration.
So we define $p^{\circ}_k(n)=p(...p(n))$ ( $p$ applied $k$ times successively).
For, for example $12$, we have $p(12)=16$ and $p(16)=15$ and $p(15)=9$ and $p(9)=4$ and $p(4)=3$ and $p(3)=1$, so we have $p^{\circ}_l(12))=1$ for every $l\geq 6$.
If we take some number and apply $p$ to that number successively then two types of behavior can arise:
1) we run into a loop of length $\geq 1$
2) we escape into an infinity
Because looping seems hard to detect, I would like to know something more about 2).
At first, a question naturally arises:
Is there some natural number for which successive appliance of a function $p$ never loops? That is, phrased differently, is there some $n \in \mathbb N$ such that we have that $\{p(n),p^{\circ}_2,p^{\circ}_3,...\}$ is unbounded?
 A: The Catalan conjecture (sometimes called the Catalan–Dickson conjecture), is that every aliquot sequence ends either at $1$ or in a finite loop (reaching a perfect, amicable, or sociable number).
It is currently an open question.  For example this talk from 2016 by A. S. Mosunov, "What do we know about aliquot sequences?", explains the conjecture of Guy and Selfridge that an unbounded aliquot sequence exists:

We do not know any $n$ such that the aliquot sequence starting
  at $n$ is unbounded. However, up to $1000$ there are $12$ possible candidates:
  $276,306,396,552,564,660,696,780,828,888,966,996$. The aliquot sequences starting at $276,552,564,660$ and $966$ were studied by Derrik Lehmer.

These latter cases are known as the Lehmer Five.

Guy-Selfridge Counter Conjecture. There are infinitely
  many aliquot sequences that are unbounded.
Guy-Selfridge Heuristics. Most of the aliquot sequences
  starting with even number are unbounded, while most of the
  aliquot sequences starting with an odd number are bounded.

This is a pretty remarkable claim, but see Guy and Selfridge (1975) What drives an aliquot sequence? for some reasoning.
