Divisor Sum of 45

Example of Divisor Sum of Integer

$\map {\sigma_1} {45} = 78$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$45 = 3^2 \times 5$

Hence:

\(\ds \map {\sigma_1} {45}\)

\(=\)

\(\ds \frac {3^3 - 1} {3 - 1} \times \frac {5^2 - 1} {5 - 1}\)

Divisor Sum of Integer

\(\ds \)

\(=\)

\(\ds \frac {27 - 1} 2 \times \frac {25 - 1} 4\)

\(\ds \)

\(=\)

\(\ds 13 \times 6\)

\(\ds \)

\(=\)

\(\ds 78\)

$\blacksquare$