OPEN
This is open, and cannot be resolved with a finite computation.
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 integers in $A$ are also known as primitive pseudoperfect numbers and are listed as
A006036 in the OEIS.
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 (which are listed as
A119425 in the OEIS).
Benkoski and Erdős
[BeEr74] ask about these two sets, and also about the set of $n$ that have a divisor expressible as a distinct sum of other divisors of $n$, but where no proper divisor of $n$ has this property.
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This page was last edited 19 October 2025.
Additional thanks to: Zachary Chase and Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #469, https://www.erdosproblems.com/469, accessed 2026-01-16
