Divisor Sum of 30,240

Example of Divisor Sum of Integer

$\map {\sigma_1} {30 \, 240} = 120 \, 960$

where $\sigma_1$ denotes the divisor sum function.

We have that:

$30 \, 240 = 2^5 \times 3^3 \times 5 \times 7$

Hence:

$\ds \map {\sigma_1} {30 \, 240}$

$=$

$\ds \frac {2^6 - 1} {2 - 1} \times \frac {3^4 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1}$

$\ds $

$=$

$\ds 63 \times 40 \times 6 \times 8$

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds 4 \times \paren {2^5 \times 3^3 \times 5 \times 7}$

$\ds $

$=$

$\ds 120 \, 960$

$\blacksquare$