Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/630

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

The number $630$ has the property that:

$\map {\sigma_0} {630} \divides \map \phi {630} \divides \map {\sigma_1} {630}$

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

\(\ds \map {\sigma_0} {630}\)

\(=\)

\(\, \ds 24 \, \)

\(\ds \)

$\sigma_0$ of $630$

\(\ds \map \phi {630}\)

\(=\)

\(\, \ds 144 \, \)

\(\, \ds = \, \)

\(\ds 6 \times 24\)

$\phi$ of $630$

\(\ds \map {\sigma_1} {630}\)

\(=\)

\(\, \ds 1872 \, \)

\(\, \ds = \, \)

\(\ds 13 \times 144\)

$\sigma_1$ of $630$

$\blacksquare$