Erdős Problem 285

Reference: erdosproblems.com/285

theorem erdos_285 (f : ℕ → ℕ) (S : Set ℕ) (hS : S = {k : ℕ | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i : Fin k.succ, 1 / ↑(n i)}) (h : ∀ k ∈ S, IsLeast {x : ℕ | ∃ (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i : Fin k.succ, 1 / ↑(n i)), n (Fin.last k) = x} (f k)) : (∃ (o : ℕ → ℝ) (_ : Filter.Tendsto o Filter.atTop (nhds 0)), ∀ k ∈ S, ↑(f k) = (1 + o k) * Real.exp 1 / (Real.exp 1 - 1) * (↑k + 1)) ↔ True

Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \cdots < n_k$ with $$ 1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}. $$ Is it true that $$ f(k) = (1 + o(1)) \frac{e}{e - 1} k ? $$

Proved by Martin [Ma00].

[Ma00] Martin, Greg, Denser Egyptian fractions. Acta Arith. (2000), 231-260.

theorem erdos_285.variants.lb (f : ℕ → ℕ) (S : Set ℕ) (hS : S = {k : ℕ | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i : Fin k.succ, 1 / ↑(n i)}) (h : ∀ k ∈ S, IsLeast {x : ℕ | ∃ (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i : Fin k.succ, 1 / ↑(n i)), n (Fin.last k) = x} (f k)) : ∃ (o : ℕ → ℝ) (_ : Filter.Tendsto o Filter.atTop (nhds 0)), ∀ k ∈ S, (1 + o k) * Real.exp 1 / (Real.exp 1 - 1) * (↑k + 1) ≤ ↑(f k)

It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$.