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on Erdős #269 · irrationality

The statement

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$ . Is the sum

\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},

where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$ , irrational?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.