Sigma Function of 5564

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Example of Sigma Function of Integer

$\map \sigma {5564} = 10 \, 584$

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


Proof

From Sigma Function of Integer

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

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


We have that:

$5564 = 2^2 \times 13 \times 107$


Hence:

\(\displaystyle \map \sigma {5020}\) \(=\) \(\displaystyle \frac {2^3 - 1} {2 - 1} \times \paren {13 + 1} \times \paren {107 + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 7 1 \times 14 \times 108\)
\(\displaystyle \) \(=\) \(\displaystyle 7 \times \paren {2 \times 7} \times \paren {2^2 \times 3^3}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^3 \times 7^2\)
\(\displaystyle \) \(=\) \(\displaystyle 10 \, 584\)

$\blacksquare$