Derived sequences

We define a multiplicative arithmetic function D by assigning D(p a) = apa-1, when p is a prime and a is a positive integer, and, for n ≥ 1, we set D0(n) = n and Dk(n) = D(D k-1(n)) when k ≥ 1. We term {Dk(n)}k=0∞ the derived sequence of n. We show that all derived sequences of n < 1.5 · 1010 are bounded, and that the density of those n ∈ ℕ with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences. Known bounded derived sequences end (effectively) in cycles of lengths only 1 to 6, and 8, yet the existence of cycles of arbitrary length is conjectured. We prove the existence of derived sequences of arbitrarily many terms without a cycle