  In this note we study numbers which occur as conductors of elliptic curves
over Q. We show, by constructing families of elliptic curves with quadratic
discriminant and invoking a theorem of Iwaniec, that this set contains
infinitely many almost primes. We show, assuming a strong version of the
Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly
bounded density in the set of primes. Studying the Cremona and Stein-Watkins
databases of elliptic curves we conjecture that the set of conductors should be
of density zero in the set of natural numbers and that the set of prime
conductors should be of density zero in the set of prime numbers.
