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The Arithmetic Partial Derivative
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Brad Emmons and Xiao Xiao

Department of Mathematics

Utica University

Utica, NY 13502

USA

**Abstract:**

The arithmetic partial derivative (with respect to a prime *p*) is a
function from the set of integers that sends *p* to 1 and satisfies
the Leibniz rule. In this paper, we prove that the *p*-adic valuation
of the sequence of higher order partial derivatives is eventually
periodic. We also prove a criterion to determine when an integer has
integral anti-partial derivatives. As an application, we show that
there are infinitely many integers with exactly *n* integral anti-partial
derivatives for every nonnegative integer *n*.

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(Concerned with sequences
A003415.)

Received January 29 2022; revised version received June 1 2022.
Published in *Journal of Integer Sequences*,
June 4 2022.

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