TITLE: Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and  $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?
QUESTION [25 upvotes]: Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and
$$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$
See also https://math.stackexchange.com/questions/42624/double-sum-miklos-schweitzer-2010

REPLY [5 votes]: In the recent issue of Matematikai Lapok, the report on the Schweitzer contest attributes the problem to Zoltan Buczolich and Julien Bremont.