Problem Statement
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.
Is it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\limsup f(n)/n=\infty$?
Is it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\limsup f(n)/n=\infty$?
Categories:
Number Theory Divisors
Progress
A question of Erdős reported in problem B2 of Guy's collection [Gu04]. The function $f(n)$ is undefined for $n=2$ and $n=5$, but is likely well-defined for all $n\geq 6$ (which would follow from a strong form of Goldbach's conjecture).The sequence of values of $f(n)$ is given by A167485 in the OEIS.
See also [468].
The strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to [468], in which he proves that the upper density of $\{ n : f(n)\leq \delta n\}$ is $\ll \delta^2$.
Source: erdosproblems.com/1054 | Last verified: January 19, 2026