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Volume 10, Issue 1


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Ordered and Unordered Factorizations of Integers
Arnold Knopfmacher
Michael Mays

We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these methods in Mathematica and compare the speeds of various approaches to generating these factorizations in practice.





*Ordered Factorizations

*Unordered Factorizations

*Highly Factorable Numbers

*Factorizations with Relatively Prime Parts




About the Authors
Arnold Knopfmacher is a Professor of Mathematics at the University of the Witwatersrand, Johannesburg, where he also obtained his Ph.D. degree. His main research interests are in enumerative combinatorics and elementary number theory. He is Director of The John Knopfmacher Centre for Applicable Analysis and Number Theory, which was established in 1992 by his late father, a distinguished number theorist. Prior to John's death in 1999, Arnold and John collaborated extensively producing over thirty joint papers together.

Michael Mays is a Professor of Mathematics at West Virginia University, where he does research in combinatorics and number theory and develops software and course materials for the Institute for Math Learning. He has maintained his collaboration with Arnold Knopfmacher with five visits over the last decade to The John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand.

Arnold Knopfmacher
The John Knopfmacher Centre for Applicable Analysis and Number Theory
University of the Witwatersrand
Johannesburg 2050, South Africa

Michael Mays
Department of Mathematics
West Virginia University
Morgantown, WV 26506-6310

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