Let $u=(u_n)_{n\ge0}$ be a strictly increasing sequence of natural integers.
Are there known (and reasonably explicit) formulas for some series of the form :
$$S_u=\sum_{n=0}^\infty\frac{1}{(u_n)!}$$
?
Of course, I put aside the well known case where : $\exists p\in\mathbb{N}^\star;\forall n\in\mathbb{N},u_n=pn$.
For example, can we expect some formulas for :
$$\sigma=\sum_{n=0}^\infty\frac{1}{(n^2)!}$$ $$\tau=\sum_{n=0}^\infty\frac{1}{(n!)!}$$
?
You did not yet formally rule out all the easy cases, like $$ \sum_{n=1}^\infty \frac{1}{(2n+1)!} = \frac{e-e^{-1}}{2} $$
So to get a hard case, you must rule out a subset $A$ of $\mathbb N$ that is eventually periodic.