Problem 1054

Erdős Problem #1054: declared status 'open'. Formalized: yes. 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 $\limsup f(n)/n=\infty$? Current best: 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). Prize: no. OEIS: A167485. Tags: divisors, number theory.