Divisor Sum of 240

Example of Divisor Sum of Integer

$\map {\sigma_1} {240} = 744$

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:

$240 = 2^4 \times 3 \times 5$

Hence:

$\ds \map {\sigma_1} {240}$

$=$

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

$\ds $

$=$

$\ds \frac {31} 1 \times \frac 8 2 \times \frac {24} 4$

$\ds $

$=$

$\ds 31 \times 4 \times 6$

$\ds $

$=$

$\ds 744$

$\blacksquare$