Journal of Integer Sequences, Vol. 6 (2003), Article 03.2.7

Further Results on Derived Sequences

Kevin G. Hare and Soroosh Yazdani
Department of Mathematics
970 Evans Hall
University of California
Berkeley, CA 94720-3840
USA

Abstract: In 2003 Cohen and Iannucci introduced a multiplicative arithmetic function D by assigning D(p^a) = a p^{a-1} when p is a prime and a is a positive integer. They defined D^0(n) = n and D^k(n) = D(D^{k-1}(n)) and they called (D^k(n)), k >= 0 the derived sequence of n. This paper answers some open questions about the function D and its iterates. We show how to construct derived sequences of arbitrary cycle size, and we give examples for cycles of lengths up to 10. Given n, we give a method for computing m such that D(m)=n, up to a square free unitary factor.

Received April 21, 2003; revised version received June 23, 2003. Published in Journal of Integer Sequences July 9, 2003.