Sigma Function of 6232

From ProofWiki
(Redirected from Sigma of 6232)
Jump to navigation Jump to search

Example of Sigma Function of Integer

$\sigma \left({6232}\right) = 12 \, 600$

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:

$6232 = 2^3 \times 19 \times 41$


Hence:

\(\displaystyle \sigma \left({6232}\right)\) \(=\) \(\displaystyle \frac {2^4 - 1} {2 - 1} \times \left({19 + 1}\right) \times \left({41 + 1}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {15} 1 \times 20 \times 42\)
\(\displaystyle \) \(=\) \(\displaystyle \left({3 \times 5}\right) \times \left({2^2 \times 5}\right) \times \left({2 \times 3 \times 7}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^2 \times 5^2 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle 12 \, 600\)

$\blacksquare$