AMC 12 2022a Problem 23

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 $1, 2, 3, \ldots, n$. For how many integers with $1\le{n}\le{22}$ is $k_n<L_n$?

$\textbf{(A) }0 \textbf{(B) }3 \textbf{(C) }7 \textbf{(D) }8 \textbf{(E) }10$