##
**
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.

**
Full version: pdf,
dvi,
ps,
latex
**

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

Return to
**Journal of Integer Sequences home page**