Erdős Problem 825

Reference: erdosproblems.com/825

theorem erdos_825 : (∃ (C : ℝ) (_ : C > 0), ∀ (n : ℕ), ↑((ArithmeticFunction.sigma 1) n) > C * ↑n → ∃ s ⊆ n.properDivisors, n = s.sum id) ↔ sorry

Is there an absolute constant $C > 0$ such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$?

theorem erdos_825.variants.necessary_cond (C : ℝ) (hC : 0 < C) (h : ∀ (n : ℕ), ↑((ArithmeticFunction.sigma 1) n) > C * ↑n → ∃ s ⊆ n.properDivisors, n = s.sum id) : 2 < C

Show that if the constant $C > 0$ is such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$, then we must have $C > 2$.