Divisor Sum of 5020

Example of Divisor Sum of Integer

$\map {\sigma_1} {5020} = 10 \, 584$

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:

$5020 = 2^2 \times 5 \times 251$

Hence:

$\ds \map {\sigma_1} {5020}$

$=$

$\ds \frac {2^3 - 1} {2 - 1} \times \paren {5 + 1} \times \paren {251 + 1}$

$\ds $

$=$

$\ds \frac 7 1 \times 6 \times 252$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^3 \times 3^3 \times 7^2$

$\ds $

$=$

$\ds 10 \, 584$

$\blacksquare$