Divisor Sum of Integer/Examples/20

Example of Divisor Sum of Integer

$\map {\sigma_1} {20} = 42$

where $\sigma_1$ denotes the divisor sum function.

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

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

We have that:

$20 = 2^2 \times 5$

Hence:

$\ds \map {\sigma_1} {20}$

$=$

$\ds \frac {2^3 - 1} {2 - 1} \times \frac {5^2 - 1} {5 - 1}$

$\ds $

$=$

$\ds \frac 7 1 \times \frac {6 \times 4} 4$

$\ds $

$=$

$\ds 7 \times 6$

$\ds $

$=$

$\ds 42$

$\blacksquare$