Consequences in number theory if we have ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k>1$?

Question

I asked this question $4$ years ago in MO , but There is no affirmative answer about existence of such integer $k>1$ for which ${\sigma}^{k}(x)\equiv 0\pmod{x}$ hold , My question here : What are the consequence of this claim if it would be hold ?

Note:$${\sigma}^{k}(x )=\sigma(\sigma(\sigma(\sigma(\cdots x))))$$