2021 AMC 12A Problem 25

Let d(n) denote the number of positive integers that divide n, including 1 and n. For example, d(1)=1,d(2)=2, and d(12)=6. (This function is known as the divisor function.) Letf(n)=\frac{d(n)}{\sqrt [3]n}.There is a unique positive integer N such that f(N)>f(n) for all positive integers n\ne N. What is the sum of the digits of N?

\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9

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