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set of positve integers with strange conditions

Source: 2019 China TST Test 3 P2

March 23, 2019
Divisorsnumber theory

Problem Statement

Let SS be a set of positive integers, such that n∈Sn \in S if and only if ∑d∣n,d<n,d∈Sd≤n\sum_{d|n,d<n,d \in S} d \le n Find all positive integers n=2k⋅pn=2^k \cdot p where kk is a non-negative integer and pp is an odd prime, such that ∑d∣n,d<n,d∈Sd=n\sum_{d|n,d<n,d \in S} d = n
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