Summary: We define a multiplicative arithmetic function $D$ by assigning $D(p^a)=$ap^a-1, when $p$ is a prime and a is a positive integer, and, for $n >= 1$, we set D^$0(n)=n$ and $D^k(n)=$D(D^k-1$(n))$ when $k>= 1$. We term ${D^k(n)}$_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 $n$ in 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.

11Y55, 11A25, 11B83

arithmetic functions, density, unbounded sequences, cycles