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

## Derived Sequences

### G. L. Cohen 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.

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