KöMaL 2024 B5419

Let $q(n)$ denote the greatest odd divisor of positive integer $n$. Let $P(n)=q(1)+q(2)+\ldots+q(n)$ and $S(n)=1+2+\ldots+n$. Prove that the ratio $P(n)/S(n)$ is smaller than $2/3$ for infinitely many values of $n$, and also greater than $2/3$ for infinitely many values of $n$.