Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Question

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$?

Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of divisors function.

Note : I know only it is for $k=1$ and $k=2$

Thank you for any help .