Looking for some references or hints on the following:
Suppose we have a strictly increasing sequence of positive integers $\alpha=\{a_k\}_{k=1}^{n}$ and we define the sum $$ S_\alpha=\sum_{k=1}^n\frac{1}{a_k^2}. $$ Clearly when $\alpha$ has a finite number of terms, i.e. when $n<\infty$, then $S_\alpha$ is rational. And when $\alpha=\mathbb{N}$ we have $S_\alpha=\frac{\pi^2}{6}$, which is irrational.
I am interested in finding infinite sequences for which $S_\alpha$ is rational, if there are any. Does anyone know of a good source where this idea might be discussed?