Product of Divisor Sum and Euler Phi Functions

Theorem
Let $n$ be an integer such that $n \ge 2$.

Let the prime decomposition of $n$ be:
 * $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

Let $\phi \left({n}\right)$ be the Euler phi function of $n$.

Then:
 * $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = n^2 \prod_{1 \mathop \le i \mathop \le r} \left({1 - \frac 1 {p_i^{k_i + 1}}}\right)$

Proof
From Euler Phi Function of Integer:
 * $\displaystyle \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i - 1} \left({p_i - 1}\right)$

From Sigma Function of Integer:
 * $\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

So:
 * $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \left({\frac {p_i^{k_i + 1} - 1} {p_i - 1}}\right) p_i^{k_i - 1} \left({p_i - 1}\right)$

Taking a general factor of this product:

So:
 * $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i^{2 k_i} \left({1 - \frac 1 {p_i^{k_i + 1} } }\right)$

Hence:
 * $\displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{2 k_i} = \left({\prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}}\right)^2 = n^2$

and the result follows.