Divisor Sum of 14,288

Example of Divisor Sum of Integer

$\map {\sigma_1} {14 \, 288} = 29 \, 760$

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:

$14 \, 288 = 2^4 \times 19 \times 47$

Hence:

$\ds \map {\sigma_1} {14 \, 288}$

$=$

$\ds \frac {2^5 - 1} {2 - 1} \times \frac {19^2 - 1} {19 - 1} \times \frac {47^2 - 1} {47 - 1}$

$\ds $

$=$

$\ds \frac {31} 1 \times \frac {20 \times 18} {18} \times \frac {48 \times 46} {46}$

$\ds $

$=$

$\ds 31 \times 20 \times 48$

$\ds $

$=$

$\ds 31 \times \paren {2^2 \times 5} \times \paren {2^4 \times 3}$

$\ds $

$=$

$\ds 2^6 \times 3 \times 5 \times 31$

$\ds $

$=$

$\ds 29 \, 760$

$\blacksquare$