Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for $f$ if and only if, $x$ is prime.
If we iterate starting at any integer $x$ we get a dynamical system.
Computations with Maple showed that for all integer $x$ in $[2,2000000]$ there exists an integer $n$ such that $(f \circ f \circ \dots \circ f)(x)$ is prime; that is for each integer $x$ the sequence generated is stationary.
The question is: can anyone help proving the conjecture ?
Thank you for any help on the subject.
Best Answer
$\sigma(n)$ grows not too much faster than $n$: $\sigma(n) = O(n \log \log n)$. This is slow enough that we should have $\log f^{(n)}(x) = O(n \log n)$ say. Heuristically, the probability that $f^{(n)}(x)$ is prime should be something like $1/\log f^{(n)}(x)$, and since $\sum_n \dfrac{1}{n \log n}$ diverges, we would expect to eventually reach a prime with probability $1$. Of course, this is not a proof, but it does provide some justification for believing the conjecture.