Problem

Source: 2019 China TST Test 3 P2

Tags: Divisors, number theory



Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$