Divisor Sum of 31,185

Example of Divisor Sum of Integer

$\map {\sigma_1} {31 \, 185} = 69 \, 696$

where $\sigma_1$ denotes the divisor sum function.

We have that:

$31 \, 185 = 3^4 \times 5 \times 7 \times 11$

Hence:

$\ds \map {\sigma_1} {31 \, 185}$

$=$

$\ds \dfrac {3^5 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1} \times \paren {11 + 1}$

$\ds $

$=$

$\ds \dfrac {242} 2 \times 6 \times 8 \times 12$

$\ds $

$=$

$\ds 121 \times 6 \times 8 \times 12$

$\ds $

$=$

$\ds 11^2 \times \paren {2 \times 3} \times 2^3 \times \paren {2^2 \times 3}$

$\ds $

$=$

$\ds 2^6 \times 3^2 \times 11^2$

$\ds $

$=$

$\ds 69 \, 696$

$\blacksquare$