Divisor Sum of Integer/Examples/20
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Example of Divisor Sum of Integer
- $\map {\sigma_1} {20} = 42$
where $\sigma_1$ denotes the divisor sum function.
Proof
From Divisor Sum of Integer:
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
- $20 = 2^2 \times 5$
Hence:
\(\ds \map {\sigma_1} {20}\) | \(=\) | \(\ds \frac {2^3 - 1} {2 - 1} \times \frac {5^2 - 1} {5 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 7 1 \times \frac {6 \times 4} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \times 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 42\) |
$\blacksquare$