This question related to [this] 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.


[this]:https://math.stackexchange.com/q/1357465/230303