Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.2

Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?

Pentti Haukkanen, Mika Mattila, and Jorma K. Merikoski
School of Information Sciences
FI-33014 University of Tampere

Timo Tossavainen
School of Applied Educational Science and Teacher Education
University of Eastern Finland
P.O.Box 86, FI-57101 Savonlinna


Given $n\in\mathbb{Z} $, its arithmetic derivative n' is defined as follows: (i)  0'=1'=(-1)'=0. (ii) If $n=up_1\cdots p_k$, where $u=\pm 1$ and $p_1,\dots,p_k$ are primes (some of them possibly equal), then

\begin{displaymath}n'=n\sum_{j=1}^k\frac{1}{p_j}=u\sum_{j=1}^kp_1\cdots p_{j-1}p_{j+1}\cdots p_k.

An analogous definition can be given in any unique factorization domain. What about the converse? Can the arithmetic derivative be (well-)defined on a non-unique factorization domain? In the general case, this remains to be seen, but we answer the question negatively for the integers of certain quadratic fields. We also give a sufficient condition under which the answer is negative.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000040 A003415 A005117.)

Received October 30 2012; revised version received January 1 2013. Published in Journal of Integer Sequences, January 1 2013.

Return to Journal of Integer Sequences home page