2022 AMC 12A 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 \qquad\textbf{(B) }3 \qquad\textbf{(C) }7 \qquad\textbf{(D) }8\qquad\textbf{(E) }10

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.