Divisor Sum of 497

Example of Divisor Sum of Integer

$\map {\sigma_1} {497} = 576$

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:

$497 = 7 \times 71$

Hence:

$\ds \map {\sigma_1} {497}$

$=$

$\ds \paren {7 + 1} \paren {71 + 1}$

$\ds $

$=$

$\ds 8 \times 72$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^6 \times 3^2$

$\ds $

$=$

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

$\ds $

$=$

$\ds 24^2$

$\ds $

$=$

$\ds 576$

$\blacksquare$