TITLE: Double sum - Miklos Schweitzer 2010
QUESTION [50 upvotes]: There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is:

Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty $ and $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$

REPLY [18 votes]: Prof. Noam Elkies has given a answer at Mathoverflow. Here is his answer.