Sigma Function of 1,175,265

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Example of Sigma Function of Integer

$\map \sigma {1 \, 175 \, 265} = 2 \, 614 \, 248$

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


Proof

We have that:

$1 \, 175 \, 265 = 3^2 \times 5 \times 7^2 \times 13 \times 41$


Hence from Sigma Function of Integer:

\(\ds \map \sigma {1 \, 175 \, 265}\) \(=\) \(\ds \frac {3^3 - 1} {3 - 1} \times \paren {5 + 1} \times \frac {7^3 - 1} {7 - 1} \times \paren {13 + 1} \times \paren {41 + 1}\)
\(\ds \) \(=\) \(\ds \frac {26} 2 \times 6 \times \frac {342} 6 \times 14 \times 42\)
\(\ds \) \(=\) \(\ds 13 \times 6 \times 57 \times 14 \times 42\)
\(\ds \) \(=\) \(\ds 13 \times \paren {2 \times 3} \times \paren {3 \times 19} \times \paren {2 \times 7} \times \paren {2 \times 3 \times 7}\)
\(\ds \) \(=\) \(\ds 2^3 \times 3^3 \times 7^2 \times 13 \times 19\)
\(\ds \) \(=\) \(\ds 2 \, 614 \, 248\)

$\blacksquare$