Divisor Sum of Non-Square Semiprime/Examples/14

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Example of Divisor Sum of Non-Square Semiprime

$\map {\sigma_1} {14} = 24$

where $\sigma_1$ denotes the divisor sum.


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:

$14 = 2 \times 7$

Hence:

\(\ds \map {\sigma_1} {14}\) \(=\) \(\ds \frac {2^2 - 1} {2 - 1} \times \frac {7^2 - 1} {7 - 1}\)
\(\ds \) \(=\) \(\ds \frac 3 1 \times \frac {48} 6\)
\(\ds \) \(=\) \(\ds 3 \times 8\)
\(\ds \) \(=\) \(\ds 24\)

$\blacksquare$


Proof 2

We have that:

$14 = 2 \times 7$

and so by definition is a semiprime whose prime factors are distinct.


Hence:

\(\ds \map {\sigma_1} {14}\) \(=\) \(\ds \paren {2 + 1} \paren {7 + 1}\) Divisor Sum of Non-Square Semiprime
\(\ds \) \(=\) \(\ds 3 \times 8\)
\(\ds \) \(=\) \(\ds 24\)

$\blacksquare$