On the sum of iterations of the Euler function

We study the sum F(n) =∑ k=1 κ(n) φ^(k)(n). of consecutive iterations of the Euler function φ(n) (where the last iteration satisfies φ^(κ(n))(n) = 1). We show that for almost all n, the difference |F(n) - n is not too small, and the ratio n/F(n) is not an integer. The latter result is related to a question about the so-called perfect totient numbers, for which F(n) = n.