FormalConjectures.ErdosProblems.«410»

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import FormalConjecturesUtil

Erdős Problem 410

Reference: erdosproblems.com/410


open ArithmeticFunction Filter


namespace Erdos410

Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.

Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$?

This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_410 : answer(sorry) ↔ ∀ n > 1,
    Tendsto (fun k : ℕ ↦ ((sigma 1)^[k] n : ℝ) ^ (1 / (k : ℝ))) atTop atTop := by⊢ True ↔ ∀ n > 1, Tendsto (fun k => ↑((⇑(sigma 1))^[k] n) ^ (1 / ↑k)) atTop atTop
  sorryAll goals completed! 🐙


end Erdos410