Divisor Sum of 418

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Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {418} = 720$

where $\sigma_1$ denotes the divisor sum function.


Proof

We have that:

$418 = 2 \times 11 \times 19$


Hence:

\(\ds \map {\sigma_1} {418}\) \(=\) \(\ds \paren {2 + 1} \paren {11 + 1} \paren {19 + 1}\) Divisor Sum of Square-Free Integer
\(\ds \) \(=\) \(\ds 3 \times 12 \times 20\)
\(\ds \) \(=\) \(\ds 3 \times \paren {2^2 \times 3} \times \paren {2^2 \times 5}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2 \times 5\)
\(\ds \) \(=\) \(\ds 12^3\)
\(\ds \) \(=\) \(\ds 720\)

$\blacksquare$