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

School of Information Sciences

FI-33014 University of Tampere

Finland

Timo Tossavainen

School of Applied Educational Science and Teacher Education

University of Eastern Finland

P.O.Box 86, FI-57101 Savonlinna

Finland

**Abstract:**

Given
,
its arithmetic derivative *n*' is defined as follows: (i)
0'=1'=(-1)'=0. (ii) If
,
where
and
are primes (some of them possibly equal), then

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.

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.

(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.

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