  Let E be a non-CM elliptic curve defined over Q. For each prime p of good
reduction, E reduces to a curve E_p over the finite field F_p. For a given
squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p),
whose values are associated with the reduction of E over F_p. We are
particularly interested in two sequences: f_p(E) =p + 1 - a_p(E) and f_p(E) =
a_p(E)^2 - 4p. We present two results towards the goal of determining how often
the values in a given sequence are squarefree. First, for any fixed curve E, we
give an upper bound for the number of primes p up to X for which f_p(E) is
squarefree. Moreover, we show that the conjectural asymptotic for the prime
counting function \pi_{E,f}^{SF}(X) := #{p \leq X: f_p(E) is squarefree} is
consistent with the asymptotic for the average over curves E in a suitable box.
