Sigma Function of 1184

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Example of Sigma Function of Integer

$\sigma \left({1184}\right) = 2394$

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


Proof

From Sigma Function of Integer

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

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


We have that:

$1184 = 2^5 \times 37$

Hence:

\(\displaystyle \sigma \left({1184}\right)\) \(=\) \(\displaystyle \frac {2^6 - 1} {2 - 1} \times \frac {37^2 - 1} {37 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {63} 1 \times \frac {38 \times 36} {36}\)
\(\displaystyle \) \(=\) \(\displaystyle 63 \times 38\)
\(\displaystyle \) \(=\) \(\displaystyle \left({3^2 \times 7}\right) \times \left({2 \times 19}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3^2 \times 7 \times 19\)
\(\displaystyle \) \(=\) \(\displaystyle 2394\)

$\blacksquare$