Divisor Sum of 12,496

Example of Divisor Sum of Integer

$\map {\sigma_1} {12 \, 496} = 26 \, 784$

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:

$12 \, 496 = 2^4 \times 11 \times 71$

Hence:

$\ds \map {\sigma_1} {12 \, 496}$

$=$

$\ds \frac {2^5 - 1} {2 - 1} \times \frac {11^2 - 1} {11 - 1} \times \frac {71^2 - 1} {71 - 1}$

$\ds $

$=$

$\ds \frac {31} 1 \times \frac {12 \times 10} {10} \times \frac {72 \times 70} {70}$

$\ds $

$=$

$\ds 31 \times 12 \times 72$

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds 26 \, 784$

$\blacksquare$