Divisor Sum of 168

Example of Divisor Sum of Integer

$\map {\sigma_1} {168} = 480$

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:

$168 = 2^3 \times 3 \times 7$

Hence:

$\ds \map {\sigma_1} {168}$

$=$

$\ds \frac {2^4 - 1} {2 - 1} \times \frac {3^2 - 1} {3 - 1} \times \frac {7^2 - 1} {7 - 1}$

$\ds $

$=$

$\ds \frac {15} 1 \times \frac {4 \times 2} 2 \times \frac {8 \times 6} 6$

$\ds $

$=$

$\ds 15 \times 4 \times 8$

$\ds $

$=$

$\ds \paren {3 \times 5} 2^2 \times 2^3$

$\ds $

$=$

$\ds 2^5 \times 3 \times 5$

$\ds $

$=$

$\ds 480$

$\blacksquare$