erdos_277

Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$? This was answered affirmatively by Haight [Ha79].

theorem erdos_277 :
   answer(True) ↔ ∀ c : ℝ, ∃ n : ℕ, (sigma 1 n : ℝ) > c * n ∧
      ¬ ∃ (S : Finset (ℤ × ℕ)),
      (∀ z : ℤ, ∃ s ∈ S, z ≡ s.1 [ZMOD s.2]) ∧
      (∀ s ∈ S, s.2 ∣ n ∧ 1 < s.2)

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