Divisor Sum of 385

From ProofWiki
Jump to navigation Jump to search

Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {385} = 576$

where $\sigma_1$ denotes the divisor sum function.


Proof

We have that:

$385 = 5 \times 7 \times 11$


Hence:

\(\ds \map {\sigma_1} {385}\) \(=\) \(\ds \paren {5 + 1} \paren {7 + 1} \paren {11 + 1}\) Divisor Sum of Square-Free Integer
\(\ds \) \(=\) \(\ds 6 \times 8 \times 12\)
\(\ds \) \(=\) \(\ds \paren {2 \times 3} \times 2^3 \times \paren {2^2 \times 3}\)
\(\ds \) \(=\) \(\ds 2^6 \times 3^2\)
\(\ds \) \(=\) \(\ds \paren {2^3 \times 3}^2\)
\(\ds \) \(=\) \(\ds 24^2\)
\(\ds \) \(=\) \(\ds 576\)

$\blacksquare$