This question is a generalization / offshoot of this earlier MSE post:
If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, is $N$ always friendly?
Here, $\gcd(a,b)$ is the greatest common divisor of $a$ and $b$, and $\sigma(x)$ is the sum of the divisors of $x$.
Note that $$\gcd(N,\sigma(N))=\gcd(Z\sigma(Z),\sigma(Z\sigma(Z)))=\gcd(Z\sigma(Z),\sigma(Z)\sigma(\sigma(Z))) \geq \sigma(Z) > 1$$ where the last inequality follows from $1<N=Z\sigma(Z)$, so that Greening's Theorem fails to establish that $N$ is solitary. (However, this does not conclusively prove that $N$ is friendly.)
For an example: Consider $Z = 2^{p-1}$ where $p$ and $2^p - 1$ are primes.
There does not seem to be a sequence in OEIS for
Numbers $n$ such that $\gcd(n,\sigma(n))=1$ and $n\sigma(n)$ is friendly.