Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.7

The Arithmetic Partial Derivative

Brad Emmons and Xiao Xiao
Department of Mathematics
Utica University
Utica, NY 13502


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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