Numbers such that Divisor Count divides Phi divides Divisor Sum

Theorem
The sequence of integers $n$ with the property that:
 * $\map {\sigma_0} n \divides \map \phi n \divides \map \sigma n$

where:
 * $\divides$ denotes divisibility
 * $\sigma_0$ denotes the divisor counting function: the count of divisors of $n$
 * $\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$
 * $\sigma$ denotes the $\sigma$ (sigma) function: the sum of divisors of $n$

begins:
 * $1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$

Proof
By inspection and investigation.