KöMaL 2010 A520

Is there any infinite sequence $a_1$,$a_2$,... of nonnegative numbers for which $\sum_{n=1}^\infty a_n^2$ is finite and $\sum_{n=1}^\infty \bigg(\sum_{k=1}^\infty\frac{a_{kn}}{k}\bigg)^{2} = \infty?$