Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$
After some computations, I was led to the following question: let $p_n,q_n$ be two sequences of natural numbers such that $$\left|\frac{p_n}{q_n}-\frac{\pi}{2}\right|<\frac{1}{q_n^2}.$$
Can we find a subsequence of $q_n$ composed only by even number? Or is it the case that $q_n$ is odd except for a finite number of indices $n$?
Any idea or reference is welcome.