Problem

Source: Ukrainian Mathematical Olympiad 2023. Day 2, Problem 11.7

Tags: number theory



For a positive integer $n$ consider all its divisors $1 = d_1 < d_2 < \ldots < d_k = n$. For $2 \le i \le k-1$, let's call divisor $d_i$ good, if $d_{i-1}d_{i+1}$ isn't divisible by $d_i$. Find all $n$, such that the number of their good divisors is smaller than the number of their prime distinct divisors. Proposed by Mykhailo Shtandenko