Here is the MathWorld's article dedicated to the so-called Sylvester's sequence. We denote it as $e_k$, for $k\geq 0$.
Let for integers $n\geq 1$ the Möbius function $\mu(n)$, see for example this MathWorld.
I would like to improve this previous deleted question, so I think that is better ask about positive real numbers $R$ for which
$$\sum_{k=1}^\infty\frac{k^{R}\mu(k)}{\log e_k}\tag{P}$$ maybe doesn't converge. With this purpose I wondered this question.
Question (modified). Does converge $$\sum_{k=1}^\infty\frac{k^{2}\mu(k)}{\log e_k},\tag{1}$$ where $e_k$ is the Sylvester's sequence and $\mu(k)$ the Möbius function? Many thanks.
If you want to add in your answer remarks about the problem $(P)$ feel free to do it, or well in comments.