Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/140

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

The number $140$ has the property that:

$\map {\sigma_0} {140} \divides \map \phi {140} \divides \map {\sigma_1} {140}$

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

\(\ds \map {\sigma_0} {140}\)

\(=\)

\(\, \ds 12 \, \)

\(\ds \)

$\sigma_0$ of $140$

\(\ds \map \phi {140}\)

\(=\)

\(\, \ds 48 \, \)

\(\, \ds = \, \)

\(\ds 4 \times 12\)

$\phi$ of $140$

\(\ds \map {\sigma_1} {140}\)

\(=\)

\(\, \ds 336 \, \)

\(\, \ds = \, \)

\(\ds 7 \times 48\)

$\sigma_1$ of $140$

$\blacksquare$