More generally, Sándor shows that for any $n \ge 2$ there exists a finite set
$A \subseteq \mathbb{N}\setminus\{1\}$ with $\sum_{k \in A} \frac{1}{k} < n$, and no
partition into $n$ parts each of which has $\sum_{n \in A_i} \frac{1}{n} < 1$.
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theorem erdos_316.variants.generalized (n : ℕ) (hn : 2 ≤ n) : ∃ A : Finset ℕ,
A.Nonempty ∧ 0 ∉ A ∧ 1 ∉ A ∧ ∑ k ∈ A, (1 / k : ℚ) < n ∧ ∀ P : Finpartition A,
P.parts.card = n → ∃ p ∈ P.parts, 1 ≤ ∑ n ∈ p, (1 / n : ℚ)
