Abstract:
We define a multiplicative arithmetic function D by assigning D(pa) = 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(Dk-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 10 10 are bounded, and that the density of those n 2 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.
