Divisor Sum of Integer/Examples/24

Example of Divisor Sum of Integer

$\map {\sigma_1} {24} = 60$

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:

$24 = 2^3 \times 3$

Hence:

$\ds \map {\sigma_1} {24}$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds 15 \times 4$

$\ds $

$=$

$\ds 60$

$\blacksquare$