Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/168

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

The number $168$ has the property that:

$\map {\sigma_0} {168} \divides \map \phi {168} \divides \map {\sigma_1} {168}$

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

\(\ds \map {\sigma_0} {168}\)

\(=\)

\(\, \ds 16 \, \)

\(\ds \)

$\sigma_0$ of $168$

\(\ds \map \phi {168}\)

\(=\)

\(\, \ds 48 \, \)

\(\, \ds = \, \)

\(\ds 3 \times 16\)

$\phi$ of $168$

\(\ds \map {\sigma_1} {168}\)

\(=\)

\(\, \ds 480 \, \)

\(\, \ds = \, \)

\(\ds 10 \times 48\)

$\sigma_1$ of $168$

$\blacksquare$