Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/105

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

The number $105$ has the property that:

$\map {\sigma_0} {105} \divides \map \phi {105} \divides \map {\sigma_1} {105}$

where:

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


Proof

\(\ds \map {\sigma_0} {105}\) \(=\) \(\, \ds 8 \, \) \(\ds \) $\sigma_0$ of $105$
\(\ds \map \phi {105}\) \(=\) \(\, \ds 48 \, \) \(\, \ds = \, \) \(\ds 6 \times 8\) $\phi$ of $105$
\(\ds \map {\sigma_1} {105}\) \(=\) \(\, \ds 192 \, \) \(\, \ds = \, \) \(\ds 4 \times 48\) $\sigma_1$ of $105$

$\blacksquare$