I have a number theoretic question. Specifically, I am interested in knowing an elementary proof of the following theorem (I used to know it but unfortunately I forgot and can't find it anywhere):

Given $f(n) = 1 + \frac{1}{2} + \frac{1}{3} ... \frac{1}{n}$ and $2^k \leq n < 2^{k+1}$, the largest power of two that divides the reduced denominator of $f(n)$ (i.e. $b$ where $f(n) = \frac{a}{b}$ and gcd($a$,$b$) $=1$) is $2^k$.