Sigma Function of Non-Square Semiprime/Examples/115/Proof 2

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Example of Sigma Function of Non-Square Semiprime

$\sigma \left({115}\right) = 144$


Proof

We have that:

$115 = 5 \times 23$

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


Hence:

\(\displaystyle \sigma \left({115}\right)\) \(=\) \(\displaystyle \left({5 + 1}\right) \left({23 + 1}\right)\) Sigma Function of Non-Square Semiprime
\(\displaystyle \) \(=\) \(\displaystyle 6 \times 24\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 3}\right) \times \left({2^3 \times 3}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 2^4 \times 3^2\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2^2 \times 3}\right)^2\)
\(\displaystyle \) \(=\) \(\displaystyle 12^2\)
\(\displaystyle \) \(=\) \(\displaystyle 144\)

$\blacksquare$