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Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$ with $m<n$. Does \[\sum_{n\in A}\frac{1}{n}\] converge?
The same question can be asked for those $n$ which do not have distinct sums of sets of divisors, but any proper divisor of $n$ does.
Additional thanks to: Zachary Chase