Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.8

The Arithmetic Derivative and Antiderivative

Jurij Kovič
Institute of Mathematics, Physics, and Mechanics
University of Ljubljana, Slovenia


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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