The Dickson dynamical system
The map T which assigns to an integer n the sum of proper divisors
defines a dynamical system on the set of integers. It is called the
Dickson dynamical system. One can visualize it as a map on the plane,
when placing the positive integers on squares forming the Ulam spiral.
For more details, see
Oliver Knill, December 2, 2007.
Here is an animation of 100 iterates.
Fixed points of T are perfect numbers, points with period 2 are called
amicable pairs. Orbits are also called aliquot sequences.
No period 3 orbit is known, nor is it known whether there are points
which escape to infinity. Experiments indicate that there are points which
escape to infinity.
No odd fixed point is known. It is also unknown whether there are
amicable pairs, where one is even and the other is odd.