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The Arithmetic Derivative and Antiderivative
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Jurij Kovič

Institute of Mathematics, Physics, and Mechanics

University of Ljubljana, Slovenia

**Abstract:**

The notion of the arithmetic derivative, a function sending each prime
to 1 and satisfying the Leibnitz rule, is extended to the case of
complex numbers with rational real and imaginary parts. Some
constraints on the solutions to some arithmetic differential equations
are found. The homogeneous arithmetic differential equation of the *k*-th
order is studied. The factorization structure of the antiderivatives of
natural numbers is presented. Arithmetic partial derivatives are
defined and some arithmetic partial differential equations are solved.

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(Concerned with sequences
A000040
A000290
A051674.)

Received May 19 2011;
revised version received March 18 2012.
Published in *Journal of Integer Sequences*, March 25 2012.

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