Sigma Function of 220

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Example of Sigma Function of Integer

$\map \sigma {220} = 504$

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


Proof

From Sigma Function of Integer

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

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


We have that:

$220 = 2^2 \times 5 \times 11$

Hence:

\(\ds \map \sigma {220}\) \(=\) \(\ds \frac {2^3 - 1} {2 - 1} \times \frac {5^2 - 1} {5 - 1} \times \frac {11^2 - 1} {11 - 1}\)
\(\ds \) \(=\) \(\ds \frac 7 1 \times \frac {6 \times 4} 4 \times \frac {12 \times 10} {10}\)
\(\ds \) \(=\) \(\ds 7 \times 6 \times 12\)
\(\ds \) \(=\) \(\ds 7 \times \paren {2 \times 3} \times \paren {2^2 \times 3}\)
\(\ds \) \(=\) \(\ds 2^3 \times 3^2 \times 7\)
\(\ds \) \(=\) \(\ds 504\)

$\blacksquare$