Problem Statement
Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$.
Do there exist constants $c_1,c_2>0$ such that\[d_t \sim \frac{c_1}{(\log t)^{c_2}}\]as $t\to \infty$?
Do there exist constants $c_1,c_2>0$ such that\[d_t \sim \frac{c_1}{(\log t)^{c_2}}\]as $t\to \infty$?
Categories:
Number Theory Divisors
Progress
Erdős [Er70] proved that $d_t$ always exists, and that there exist some constants $c_3,c_4>0$ such that\[\frac{1}{(\log t)^{c_3}} < d_t < \frac{1}{(\log t)^{c_4}}.\]Source: erdosproblems.com/859 | Last verified: January 19, 2026