This question related to this question from SE. I'm interested to know if there exists a positive integer $x$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$ for all positive integers $k$.

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