Sigma Function of 5040

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Example of Sigma Function of Integer

$\sigma \left({5040}\right) = 19 \, 344$

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:

$5040 = 2^4 \times 3^2 \times 5 \times 7$


Hence:

\(\ds \sigma \left({5040}\right)\) \(=\) \(\ds \frac {2^5 - 1} {2 - 1} \times \frac {3^3 - 1} {3 - 1} \times \left({5 + 1}\right) \times \left({7 + 1}\right)\)
\(\ds \) \(=\) \(\ds \frac {31} 1 \times \frac {26} 2 \times 6 \times 8\)
\(\ds \) \(=\) \(\ds 31 \times 13 \times \times \left({2 \times 3}\right) \times 2^3\)
\(\ds \) \(=\) \(\ds 2^4 \times 3 \times 13 \times 31\)
\(\ds \) \(=\) \(\ds 19 \, 344\)

$\blacksquare$