Divisor Sum of 210
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Example of Divisor Sum of Square-Free Integer
- $\map {\sigma_1} {210} = 576$
where $\sigma_1$ denotes the divisor sum function.
Proof
We have that:
- $210 = 2 \times 3 \times 5 \times 7$
Hence:
\(\ds \map {\sigma_1} {210}\) | \(=\) | \(\ds \paren {2 + 1} \paren {3 + 1} \paren {5 + 1} \paren {7 + 1}\) | Divisor Sum of Square-Free Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 4 \times 6 \times 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 2^2 \times \paren {2 \times 3} \times 2^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^3 \times 3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 24^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 576\) |
$\blacksquare$