KöMaL 2021 A803

Let $\pi (n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\dots,n\}$ is called primitive if there are no two elements in it with one of them dividing the other. Prove that for $n\ge 5$ and $1\le k<\frac{\pi (n)}{2}$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements.