Sigma Function of 5564

From ProofWiki
(Redirected from Sigma of 5564)
Jump to navigation Jump to search

Example of Sigma Function of Integer

$\sigma \left({5564}\right) = 10 \, 584$

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:

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


Hence:

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

$\blacksquare$