erdos_266.variants.all_rationals:

In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing sequence $a_n$ of positive integers such that $\sum \frac{1}{a_n + t}$ converges to a rational number for all $t \in \mathbb{Q}$ such that $t \ne -a_n$ for any $n$. [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. [arXiv:2406.17593](https://arxiv.org/abs/2406.17593) (2024).

theorem erdos_266.variants.all_rationals:
    ∃ a : ℕ → ℕ, StrictMono a ∧ a 0 ≥ 1 ∧
      (∀ t : ℚ, (¬ ∃ n : ℕ, t = -(a n)) →
        (∃ q : ℚ, HasSum (fun n : ℕ => ((1 : ℝ) / ((a n) + t))) q))

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