Divisor Sum of 12,285

Example of Divisor Sum of Integer

$\map {\sigma_1} {12 \, 285} = 26 \, 880$

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:

$12 \, 285 = 3^3 \times 5 \times 7 \times 13$

Hence:

$\ds \map {\sigma_1} {12 \, 285}$

$=$

$\ds \frac {3^4 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1} \times \paren {13 + 1}$

$\ds $

$=$

$\ds 40 \times 6 \times 8 \times 14$

$\ds $

$=$

$\ds \paren {2^3 \times 5} \times \paren {2 \times 3} \times 2^3 \times \paren {2 \times 7}$

$\ds $

$=$

$\ds 2^8 \times 3 \times 5 \times 7$

$\ds $

$=$

$\ds 26 \, 880$

$\blacksquare$