The Distribution of Self-Fibonacci Divisors
  Consider the positive integers $n$ such that $n$ divides the $n$-th Fibonacci number, and their counting function $A$. We prove that \[A(x) \leq x^{1-(1/2+o(1))\log\log\log x/\log\log x}.\] 
Florian Luca, Emanuele Tron
math.NT