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.