Erdős problem 409

If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ 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

FormalConjectures/ErdosProblems/

409.lean

Retained formal statement4 of 13

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

FormalConjectures/ErdosProblems/409.leanErdos409.erdos_409.parts.i.isTheta2 linesExact file

1∀ (c : ℕ → ℕ),2  (∀ n > 0, IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} (c n)) → (fun n => ↑(c n)) =Θ[Filter.atTop] sorry

OpenStatement only, no proof