Erdős problem 269

Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$. Is the sum $$\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}$$ irrational?