Source labels openChecked July 26, 2026

Erdős Problem 409: Sigma Is Theta

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$ ?

Number theory AMS 11

Mathematical statement

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$ ?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_409.variants.sigma_isTheta

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_409.variants.sigma_isTheta (c : ℕ → ℕ)    (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :    (fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/409background