Divisor Sum of 23,625

Example of Divisor Sum of Integer

$\map {\sigma_1} {23 \, 625} = 49 \, 920$

where $\sigma_1$ denotes the divisor sum function.

We have that:

$23 \, 625 = 3^3 \times 5^3 \times 7$

Hence:

$\ds \map {\sigma_1} {23 \, 625}$

$=$

$\ds \dfrac {3^4 - 1} {3 - 1} \times \dfrac {5^4 - 1} {5 - 1} \times \paren {7 + 1}$

$\ds $

$=$

$\ds \dfrac {80} 2 \times \dfrac {624} 4 \times 8$

$\ds $

$=$

$\ds 40 \times 156 \times 8$

$\ds $

$=$

$\ds \paren {2^3 \times 5} \times \paren {2^2 \times 3 \times 13} \times 2^3$

$\ds $

$=$

$\ds 2^8 \times 3 \times 5 \times 13$

$\ds $

$=$

$\ds 49 \, 920$

$\blacksquare$