Problem Statement
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?
Categories:
Number Theory Divisors
Progress
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.
Source: erdosproblems.com/469 | Last verified: January 15, 2026