Could we predict the exact value of $\sum_{n\geq 0}\dfrac{1}{\sigma(n!)}$ while it is convergent?

Question

let $\sigma$ be sum of power divisor function , I'm interesting to evaluate the following sum $\sum_{n\geq 0}\dfrac{1}{\sigma(n!)}$ , we have that sum is converge using test creterion because we have :$ \dfrac{1}{\sigma(n!)}\leq \frac{1}{n!}$ as a reason the well known inequality :$\sigma(n) > n$ , Now I want to know if that sum can be evaluated using some theories related to its growth rate , Wolfram alpha can't give me the exact value of the titled sum however it is obviously convergent , My question here is :Could we predict the exact value of $\sum_{n\geq 0}\dfrac{1}{\sigma(n!)}$ ?