Skip to content

Erdős problem 409

Start work

If n>1n > 1 then the iteration n↦σ(n)−1n\mapsto\sigma(n) - 1 necessarily reaches a prime. Note: this is open — it is not clear that the σ iteration always terminates, since it is non-decreasing (unlike the φ iteration which is strictly decreasing).

Sources

The source's own material, retained exactly.

13 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

409.lean

Retained formal statement10 of 13

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

FormalConjectures/ErdosProblems/409.leanErdos409.erdos_409.variants.sigma_isTheta3 linesExact file
∀ (c : ℕ → ℕ),  (∀ n > 1, IsLeast {i | Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)} (c n)) →    (fun n => ↑(c n)) =Θ[Filter.atTop] sorry
OpenStatement only, no proof