Consider the "equation" \begin{equation} \frac{1}{a_n}\sum_{k=1}^n ka_k = \mathcal{O}\left(\frac{n^2}{\log n}\right).\tag{1}\label{eq:conjec} \end{equation} Does there exist some monotonically decreasing sequence $\{a_k\}$ of strictly positive real numbers which satisfies \eqref{eq:conjec}?
Of course, I would also be interested in sequences which 'do better' than that, in the sense of e.g. $\mathcal{O}(n^2/\log^2n)$ or even $\mathcal{O}(n/\log n)$, but my intuition tells me that \eqref{eq:conjec} itself cannot be satisfied given my imposed conditions on $\{a_k\}$. I am unsure how to prove this claim, though.
Also, if there does exist such a sequence, I am equally lost in how to go about determining what that sequence actually is.
I am looking mostly for hints about how to think about this or potentially fruitful methods/techniques that I am not aware of. I have been looking through tables of various closed form sums and more or less guess-and-checking modifications of them, with no avail.