Double sum – Miklos Schweitzer 2010

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 $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$


Prof. Noam Elkies has given a answer at Mathoverflow. Here is his answer.

Source : Link , Question Author : Beni Bogosel , Answer Author :
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