Abstract

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.