Divisor Sum of 110

Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {110} = 216$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$110 = 2 \times 5 \times 11$

Hence:

\(\ds \map {\sigma_1} {110}\)

\(=\)

\(\ds \paren {2 + 1} \paren {5 + 1} \paren {11 + 1}\)

Divisor Sum of Square-Free Integer

\(\ds \)

\(=\)

\(\ds 3 \times 6 \times 12\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 2^3 \times 3^3\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 216\)

$\blacksquare$