On the existence of abundant Carmichael numbers

Question

A natural number $n$ is called Carmichael if it is squarefree and for each prime factor $p$ of $n$, $p-1$ divides $n-1$. Moreover, $n$ is abundant if the sum of its divisors is $>2n$. I have checked the first Carmichael numbers and I have seen that none of them is abundant. So my question is: does there exist any abundant Carmichael number? If so, which is the smallest one?

Answer 1

See the article by Daniel Șuteu here: https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html. Abundant Carmichael numbers exist (2059832906607460252767290568443059994787898033540634712711845135488141590979778401392385 is one), but finding the smallest one is mentioned by Șuteu as an open problem.