erdos_269.variants.rational

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

theorem erdos_269.variants.rational : answer(sorry) ↔
    ∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),
    ∃ (q : ℚ), q = (series (P : Set ℕ))

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