Divisor Sum of 45,045
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Example of Divisor Sum of Integer
- $\map {\sigma_1} {45 \, 045} = 104 \, 832$
where $\sigma_1$ denotes the divisor sum function.
Proof
We have that:
- $45 \, 045 = 3^2 \times 5 \times 7 \times 11 \times 13$
Hence:
\(\ds \map {\sigma_1} {45 \, 045}\) | \(=\) | \(\ds \dfrac {3^3 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1} \times \paren {11 + 1} \times \paren {13 + 1}\) | Divisor Sum of Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {26} 2 \times 6 \times 8 \times 12 \times 14\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \times 6 \times 8 \times 12 \times 14\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \times \paren {2 \times 3} \times 2^3 \times \paren {2^2 \times 3} \times \paren {2 \times 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^7 \times 3^2 \times 7 \times 13\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 104 \, 832\) |
$\blacksquare$