Erdős Problem #40 — Growth conditions $a_n<Cn^2$ forcing unbounded $f(n)$

A sharpening of our old conjecture with Turán would state: If $a_n < Cn^2$ for all $n$ then $\limsup f(n) = \infty$. In fact, for what functions $g(n) \to \infty$ does $a_n < n^2 g(n)$ imply $\limsup f(n) = \infty$? (500 dollars)