Further results on derived sequences

Hare, Kevin G. ; Yazdani, Soroosh

Journal of integer sequences, Tome 6 (2003) no. 2

Résumé

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.

Détail

Classification : 11Y55, 11A25, 11B83
Keywords: arithmetic functions, multiplicative functions, cycles