HMMT 2002 U41

For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let

$$f(n)=\lfloor n\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{3}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor .$$

For how many values of $n, 1 \leq n \leq 100$, is $f(n)$ odd?