Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$.
A positive integer $N$ is friendly if there exists a positive integer $M \neq N$ such that $I(M) = I(N)$.
My question is the following:
Is $Z = p(p + 1) = p\sigma(p)$ always a friendly number for $p$ a prime number?
Note that $$\gcd(Z, \sigma(Z)) = \gcd(p(p + 1), (p + 1)\sigma(p + 1)) \geq p + 1 \geq 3,$$ so that Greening's Theorem fails to establish solitude.
OEIS sequence A074902 lists the following as friendly: $$6, 12, 30, 56, 132$$ but does not list $182$ and $306$, for example.