Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that
$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}=\frac{h_n}{k_n}.$
Let $L_n$ denote the least common multiple of the numbers $1,2,3,\dots ,n$. For how many integers with $1\le n\le 15$ is $k_n\ne L_n$?