Divisor Sum of Non-Square Semiprime/Examples/115
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Example of Divisor Sum of Non-Square Semiprime
- $\map {\sigma_1} {115} = 144$
where $\sigma_1$ denotes the divisor sum function.
Proof 1
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:
- $115 = 5 \times 23$
Hence:
| \(\ds \map {\sigma_1} {115}\) | \(=\) | \(\ds \frac {5^2 - 1} {5 - 1} \times \frac {23^2 - 1} {23 - 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {4 \times 6} 4 \times \frac {22 \times 24} {22}\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \times 24\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \times 3} \times \paren {2^3 \times 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^4 \times 3^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2^2 \times 3}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 12^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 144\) |
$\blacksquare$
Proof 2
We have that:
- $115 = 5 \times 23$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
| \(\ds \map {\sigma_1} {115}\) | \(=\) | \(\ds \paren {5 + 1} \paren {23 + 1}\) | Divisor Sum of Non-Square Semiprime | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \times 24\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \times 3} \times \paren {2^3 \times 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^4 \times 3^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2^2 \times 3}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 12^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 144\) |
$\blacksquare$