Problem Similar to Erdős Problem 252

Erdős Problem 252:(https://www.erdosproblems.com/252)

Let $k\geqslant 1$ and $\sigma_{k}(n):= \sum_{d|n} d^{k}$.

Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!}$ irrational?

Currently this problem is open for $k \geqslant 5$.

I am considering a different version of this problem which is as follows:

Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma(n)}{n!!}$ irrational?

I have been able to show that:

$E(m):= \displaystyle \sum_{m \geqslant 1} \frac{\sigma(2m)}{(2m)!!}$ , $O(m):= \displaystyle \sum_{m \geqslant 1} \frac{\sigma(2m-1)}{(2m-1)!!}$ are not rationals.