A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that $$\sum_{p\in\mathcal{P}} \frac1{p-1} = 1,$$ where $\mathcal{P} = \{n^a \mid n,a \ge 2\}$ is the set of powers of integers.
At the beginning, though, I thought that the sum I was looking for was $$\sum_{p\in\mathcal{P}} \frac1{p}$$ instead. The identity above tells us that the sum converges to some $S<1$.
Is $S$ known to be irrational? Is it known to be _______?
Here the blank can be filled by anything like: algebraic, transcendental, transcendental over $\mathbb{Z}[\pi]$, trascendental over $\mathbb{Z}[\pi, \log2, \log3, \dots]$...
Wikipedia says that $S\approx 0.874464368$, which makes me think that not too much is known, other than the (somewhat trivial) identity $S = \sum \mu(n)(1-\zeta(n))$...
