Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers
$a_n$ such that $\sum \frac{1}{a_n}$ converges and
$$
\liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0
$$
is not an irrationality sequence.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
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theorem erdos_264.variants.ko_tao_neg {a : ℕ → ℕ} (h₁ : StrictMono a) (h₂ : 0 ∉ Set.range a)
(h₃ : Summable ((1 : ℝ) / a ·))
(h₄ : 0 < atTop.liminf fun n ↦ a n ^ 2 * ∑' k : Set.Ioi n, (1 : ℝ) / a k ^ 2) :
¬IsIrrationalitySequence a
