Divisor Sum of 5040

Example of Divisor Sum of Integer

$\map {\sigma_1} {5040} = 19 \, 344$

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:

$5040 = 2^4 \times 3^2 \times 5 \times 7$

Hence:

$\ds \map {\sigma_1} {5040}$

$=$

$\ds \frac {2^5 - 1} {2 - 1} \times \frac {3^3 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1}$

$\ds $

$=$

$\ds \frac {31} 1 \times \frac {26} 2 \times 6 \times 8$

$\ds $

$=$

$\ds 31 \times 13 \times \times \paren {2 \times 3} \times 2^3$

$\ds $

$=$

$\ds 2^4 \times 3 \times 13 \times 31$

$\ds $

$=$

$\ds 19 \, 344$

$\blacksquare$