Divisor Sum of Non-Square Semiprime/Examples/15/Proof 1
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Example of Divisor Sum of Integer
- $\map {\sigma_1} {15} = 24$
Proof
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
- $15 = 3 \times 5$
Hence:
\(\ds \map {\sigma_1} {15}\) | \(=\) | \(\ds \frac {3^2 - 1} {3 - 1} \times \frac {5^2 - 1} {5 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 8 2 \times \frac {24} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 24\) |
$\blacksquare$