Divisor Sum of 14,536

Example of Divisor Sum of Integer

$\map {\sigma_1} {14 \, 536} = 28 \, 800$

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:

$14 \, 536 = 2^3 \times 23 \times 79$

Hence:

$\ds \map {\sigma_1} {14 \, 536}$

$=$

$\ds \frac {2^4 - 1} {2 - 1} \times \frac {23^2 - 1} {23 - 1} \times \frac {79^2 - 1} {79 - 1}$

$\ds $

$=$

$\ds \frac {15} 1 \times \frac {24 \times 22} {22} \times \frac {80 \times 78} {78}$

$\ds $

$=$

$\ds 15 \times 24 \times 80$

$\ds $

$=$

$\ds \paren {3 \times 5} \times \paren {2^3 \times 3} \times \paren {2^4 \times 5}$

$\ds $

$=$

$\ds 2^7 \times 3^2 \times 5^2$

$\ds $

$=$

$\ds 28 \, 800$

$\blacksquare$