Divisor Sum of 2400

Example of Divisor Sum of Integer

$\map {\sigma_1} {2400} = 7812$

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:

$2400 = 2^5 \times 3 \times 5^2$

Hence:

$\ds \map {\sigma_1} {2400}$

$=$

$\ds \frac {2^6 - 1} {2 - 1} \times \paren {3 + 1} \times \frac {5^3 - 1} {5 - 1}$

$\ds $

$=$

$\ds \frac {63} 1 \times 4 \times \frac {124} 4$

$\ds $

$=$

$\ds 63 \times 4 \times 31$

$\ds $

$=$

$\ds \paren {3^2 \times 7} \times 2^2 \times 31$

$\ds $

$=$

$\ds 7812$

$\blacksquare$