Divisor Sum of 180

Example of Divisor Sum of Integer

$\map {\sigma_1} {180} = 546$

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:

$180 = 2^2 \times 3^2 \times 5$

Hence:

$\ds \map {\sigma_1} {180}$

$=$

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

$\ds $

$=$

$\ds \frac 7 1 \times \frac {26} 2 \times \frac {24} 4$

$\ds $

$=$

$\ds 7 \times 13 \times 6$

$\ds $

$=$

$\ds 546$

$\blacksquare$