Divisor Sum of 714

Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {714} = 1728$

where $\sigma_1$ denotes the divisor sum function.

We have that:

$714 = 2 \times 3 \times 7 \times 17$

Hence:

$\ds \map {\sigma_1} {714}$

$=$

$\ds \paren {2 + 1} \paren {3 + 1} \paren {7 + 1} \paren {17 + 1}$

$\ds $

$=$

$\ds 3 \times 4 \times 8 \times 18$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^6 \times 3^2$

$\ds $

$=$

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

$\ds $

$=$

$\ds 12^3$

$\ds $

$=$

$\ds 1728$

$\blacksquare$