Let $a_n \to \infty$ be a monotone sequence of non-zero natural numbers.
Is $\sum_n \frac{d(n)}{(a_1 ... a_n)}$ irrational, where $d(n)$ is the number of divisors of $n$?
Solution: True (proved by Erdős and Straus, see Erdős Problems website).
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theorem erdos_258.variants.Monotone : answer(True) ↔
∀ (a : ℕ → ℤ), (∀ n, a n ≠ 0) → Monotone a →
Filter.Tendsto a Filter.atTop Filter.atTop →
Irrational (∑' (n : ℕ), ((n + 1).divisors.card / ∏ i ∈ Finset.Icc 1 n, a i))
