A self-Fibonacci number is a number $n$ such that $n$ divides $Fib(n)$.
Does this sequence contain a product of any two its elements?
I didn't found this exact result among known stuff.
I did some computer analysis and found no counterexample.
I managed to prove that if $k,l$ are in the sequence and $\gcd(k,l)=1$, then $kl$ is in the sequence.