Divisor Sum of 270

Example of Divisor Sum of Integer

$\map {\sigma_1} {270} = 720$

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:

$270 = 2 \times 3^3 \times 5$

Hence:

$\ds \map {\sigma_1} {270}$

$=$

$\ds \paren {2 + 1} \times \frac {3^4 - 1} {3 - 1} \times \paren {5 + 1}$

$\ds $

$=$

$\ds 3 \times \frac {81 - 1} 2 \times 6$

$\ds $

$=$

$\ds 3 \times 40 \times 6$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^4 \times 3^2 \times 5$

$\ds $

$=$

$\ds 720$

$\blacksquare$