Divisor Sum of 10,856

Example of Divisor Sum of Integer

$\map {\sigma_1} {10 \, 856} = 21 \, 600$

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:

$10 \, 856 = 2^3 \times 23 \times 59$

Hence:

$\ds \map {\sigma_1} {10 \, 856}$

$=$

$\ds \frac {2^4 - 1} {2 - 1} \times \paren {23 + 1} \times \paren {59 + 1}$

$\ds $

$=$

$\ds \frac {15} 1 \times 24 \times 60$

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds 21 \, 600$

$\blacksquare$