This question related to [this](https://math.stackexchange.com/q/1357465/230303) question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.


**Note.** $\sigma(x)$ is the sum of divisors of $x$, and ${\sigma}^{k}(x )=\sigma(\sigma(\sigma(\sigma(\cdots x))))$ is $\sigma$ iterated $k$ times.


**Edit.** I edited the question to avoid the trivialities.



Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?
I've decided to collect some basic observations and references for the benefit
of future readers.


A more challenging problem is to ask for integers $m$ and $p$ such that for all
integers $k$, $p\_0 = p, p\_{k+1}=\sigma(p\_k),$ and $p\_k = 0 \bmod m$. The current
problem adds the restriction that $p=m$, which implies $m$ is a multiperfect number.
Since multiperfect numbers are rare, it should be hard to find a **metaperfect**
number, a number $m$ that satisfies $\sigma^k(m) = 0 \bmod m$ for all iterations
of $\sigma$.


Indeed, $\sigma(m) \lt m\omega(m)$ for most values of $m$, so for a potentially
metaperfect number to exist, we can't depend on $\sigma(p\_k)/m$ to be coprime
to $m$ for very many $k$. More likely, $\sigma(p\_k)/m$, if integral, will share 
a small factor with $m$ and further iterations of $\sigma$ will avoid certain large 
prime factors of $m$. This is what was observed, and what I hoped to prove and 
did not in the other answer.


It is an interesting side question to determine $\min\_k g\_k$ where
$g\_0=p\_0$ and $g\_{k+1} = \gcd(p\_{k+1},g\_k)$. In particular, do the iterates of
$\sigma$ encounter a square or twice a square, regardless of the starting point?
If so, then the minimum is odd and likely 1. Otherwise $p$ is a seed for $m$, and
$p$ might be useful in looking for multiperfect numbers which are multiples of $m$.


Cohen and te Riele investigated a weaker question: Given $n$ is there a $k$ for
which $\sigma^k(n) = 0 \mod n$? They did this in a 1996 paper and asserted through
computation that the answer was yes for $n \leq 400$. Their data suggest to me both
that the weaker question has an affirmative answer, and that there are no
metaperfect numbers or even seeds for a number.


Regarding Gerry Myerson's example in a comment, I think one can construct arbitrarily
long finite sequences $p\_k$ which satisfy the congruence conditions, but to find such
$p$ with $p=m$ will result in very large values even for $k$ as small as $3$. Toward
this end, there is a 2009 paper of Katai that I have not found but think will be useful
in this study.


I recommend as a starting point for a reference search


Cohen, Graeme L., and Herman JJ te Riele. "Iterating the sum-of-divisors function." 


Experimental Mathematics 5.2 (1996): 91-100.


and (if you can get it)


Kátai, I. "On the prime power divisors of the iterates of \phi(n) and \sigma(n), Šiauliai 


Math." Semin 4.12 (2009): 125-143.


Gerhard "Not Quite A Research Announcement" Paseman, 2015.07.21


