Divisor Sum of Non-Square Semiprime/Examples/22/Proof 1

Example of Divisor Sum of Integer

$\map {\sigma_1} {22} = 36$

$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.

We have that:

$22 = 2 \times 11$

Hence:

$\ds \map {\sigma_1} {22}$

$=$

$\ds \frac {2^2 - 1} {2 - 1} \times \frac {11^2 - 1} {11 - 1}$

$\ds $

$=$

$\ds \frac 3 1 \times \frac {120} {10}$

$\ds $

$=$

$\ds 3 \times 12$

$\ds $

$=$

$\ds 3 \times \paren {2^2 \times 3}$

$\ds $

$=$

$\ds 2^2 \times 3^2$

$\ds $

$=$

$\ds \paren {2 \times 3}^2$

$\ds $

$=$

$\ds 6^2$

$\ds $

$=$

$\ds 36$

$\blacksquare$