  In this paper, we introduce the concept of $F$-perfect number, which is a
positive integer $n$ such that $\sum_{d|n,d<n}d^2=3n$. We prove that all the
$F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$
and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting
results and raise a new conjecture on perfect numbers.
