Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/30

From ProofWiki
Jump to navigation Jump to search

Examples of Numbers such that Divisor Count divides Phi divides Divisor Sum

The number $30$ has the property that:

$\map {\sigma_0} {30} \divides \map \phi {30} \divides \map {\sigma_1} {30}$

where:

$\divides$ denotes divisibility
$\sigma_0$ denotes the divisor counting function
$\phi$ denotes the Euler $\phi$ (phi) function
$\sigma_1$ denotes the divisor sum function.


Proof

\(\ds \map {\sigma_0} {30}\) \(=\) \(\, \ds 8 \, \) \(\ds \) $\sigma_0$ of $30$
\(\ds \map \phi {30}\) \(=\) \(\, \ds 8 \, \) \(\ds \) $\phi$ of $30$
\(\ds \map {\sigma_1} {30}\) \(=\) \(\, \ds 72 \, \) \(\, \ds = \, \) \(\ds 9 \times 8\) $\sigma_1$ of $30$

$\blacksquare$