Journal of Integer Sequences, Vol. 6 (2003), Article 03.1.1 |

Department of Mathematical Sciences, Faculty of Science

University of Technology, Sydney

PO Box 123, Broadway, NSW 2007

Australia

and

D. E. Iannucci

Division of Science and Mathematics

University of the Virgin Islands

St. Thomas, VI 00802

USA

**Abstract:**
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.

Received October 25, 2002;
revised version received December 1, 2002.
Published in *Journal of Integer Sequences* December 23, 2002.
Revised, February 10 2004.

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