Can a odd perfect number $n$ be a Carmichael number?
We know that all Carmichael numbers are odd and square-free.
But is there a Carmichael number that is also a perfect number?
We all know that if odd perfect numbers exists, it must be form of either $1\pmod{12}$,$81\pmod{324}$, or $117\pmod{468}$.
So if a perfect odd Carmichael number exists, then it must be $1\pmod{12}$, as $81, 117, 324$, and $468$ are all divisible by $9$ and therefore not squarefree.
And a positive integer $n$ is Carmichael number if and only $a^{n-1}\equiv 1\pmod{n}$.