Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples/3
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Examples of Numbers such that Divisor Count divides Phi divides Divisor Sum
The number $3$ has the property that:
- $\map {\sigma_0} 3 \divides \map \phi 3 \divides \map {\sigma_1} 3$
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
\(\ds \map {\sigma_0} 3\) | \(=\) | \(\, \ds 2 \, \) | \(\ds \) | $\sigma_0$ of $3$ | ||||||||||
\(\ds \map \phi 3\) | \(=\) | \(\, \ds 2 \, \) | \(\ds \) | $\phi$ of $3$ | ||||||||||
\(\ds \map {\sigma_1} 3\) | \(=\) | \(\, \ds 4 \, \) | \(\, \ds = \, \) | \(\ds 2 \times 2\) | $\sigma_1$ of $3$ |
$\blacksquare$