Divisor Sum of 1184

Example of Divisor Sum of Integer

$\map {\sigma_1} {1184} = 2394$

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:

$1184 = 2^5 \times 37$

Hence:

$\ds \map {\sigma_1} {1184}$

$=$

$\ds \frac {2^6 - 1} {2 - 1} \times \frac {37^2 - 1} {37 - 1}$

$\ds $

$=$

$\ds \frac {63} 1 \times \frac {38 \times 36} {36}$

$\ds $

$=$

$\ds 63 \times 38$

$\ds $

$=$

$\ds \paren {3^2 \times 7} \times \paren {2 \times 19}$

$\ds $

$=$

$\ds 2 \times 3^2 \times 7 \times 19$

$\ds $

$=$

$\ds 2394$

$\blacksquare$