Let $N(x)$ be the number of positive integers $n\leq x$ such that $\gcd(n,\phi(n))=1$. Here $\phi(n)$ is the Euler totient function.
A theorem of Erdos from 1948 says that

$$
N(x) \sim \frac{e^{-C_0} x}{\log\log\log x},\quad x\to \infty,
$$
where $C_0$ is the Euler constant. See also Theorem 11.23 in Montgomery-Vaughan's "Multiplicative number theory". Erdos quotes a result of Szele saying that these numbers are exactly those $n$ for which there is exactly one abstract group of order $n$.

My question is whether there is a deeper, or more enlightening, reason to count such integers?