Divisor Sum of 54

Example of Divisor Sum of Integer

$\map {\sigma_1} {54} = 120$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$54 = 2 \times 3^3$

Hence:

\(\ds \map {\sigma_1} {54}\)

\(=\)

\(\ds \frac {2^2 - 1} {2 - 1} \times \frac {3^4 - 1} {3 - 1}\)

Divisor Sum of Integer

\(\ds \)

\(=\)

\(\ds \frac {4 - 1} 1 \times \frac {81 - 1} 2\)

\(\ds \)

\(=\)

\(\ds 3 \times 40\)

\(\ds \)

\(=\)

\(\ds 3 \times \paren {2^3 \times 5}\)

\(\ds \)

\(=\)

\(\ds 120\)

$\blacksquare$