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.

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