This question related to [this] question from SE , I'm interesting to know if there is a fixed positive integer $x$  satisfing :$${\sigma}^{k}(x)\equiv 0\mod x$$ for all positive integer $k $ ,As well as i 'd like to prove or disprove it existence ?

Note(01):$\sigma(x)$ is the sum divisors of the integer $x$ and ${\sigma}^{k}(x )=\sigma(\sigma(\sigma(\sigma(\cdots x))))$

Note(02): $k$ is positive integer 
[this]:http://math.stackexchange.com/q/1357465/230303