Divisor Sum of 364

Example of Divisor Sum of Integer

$\map {\sigma_1} {364} = 784$

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:

$364 = 2^2 \times 7 \times 13$

Hence:

$\ds \map {\sigma_1} {364}$

$=$

$\ds \frac {2^3 - 1} {2 - 1} \times \frac {7^2 - 1} {7 - 1} \times \frac {13^2 - 1} {13 - 1}$

$\ds $

$=$

$\ds \frac 7 1 \times \frac {8 \times 6} 6 \times \frac {14 \times 12} {12}$

$\ds $

$=$

$\ds 7 \times 8 \times 14$

$\ds $

$=$

$\ds 7 \times 2^3 \times \paren {2 \times 7}$

$\ds $

$=$

$\ds 2^4 \times 7^2$

$\ds $

$=$

$\ds \paren {2^2 \times 7}^2$

$\ds $

$=$

$\ds 28^2$

$\ds $

$=$

$\ds 784$

$\blacksquare$