Example 26 Find $\sigma(450)=$ ?
Since $45!=2 \times 3 \times 5^{2}$, by Lemma 10 we have
$$\begin{aligned}
\sigma(450) & =\frac{2^{2}-1}{2-1} \cdot \frac{3^{3}-1}{3-1} \cdot \frac{5^{3}-1}{5-1} \\
& =3 \times 13 \times 31=1209
\end{aligned}$$