Sigma Function of Non-Square Semiprime/Examples/14

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Example of Sigma Function of Non-Square Semiprime

$\sigma \left({14}\right) = 24$

where $\sigma$ denotes the $\sigma$ function.


Proof 1

From Sigma Function of Integer

$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

where $n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.


We have that:

$14 = 2 \times 7$

Hence:

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

$\blacksquare$


Proof 2

We have that:

$14 = 2 \times 7$

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


Hence:

\(\displaystyle \sigma \left({14}\right)\) \(=\) \(\displaystyle \left({2 + 1}\right) \left({7 + 1}\right)\) Sigma Function of Non-Square Semiprime
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 8\)
\(\displaystyle \) \(=\) \(\displaystyle 24\)

$\blacksquare$