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

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 count 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 $30$

$\blacksquare$