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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the parity of the number of multiplicative partitions and related problems


Author: Paul Pollack
Journal: Proc. Amer. Math. Soc. 140 (2012), 3793-3803
MSC (2010): Primary 11N64; Secondary 11P83, 11B73
Published electronically: March 15, 2012
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Abstract: Let $ f(N)$ be the number of unordered factorizations of $ N$, where a factorization is a way of writing $ N$ as a product of integers all larger than $ 1$. For example, the factorizations of $ 30$ are

$\displaystyle 2\cdot 3\cdot 5,\quad 5\cdot 6, \quad 3\cdot 10, \quad 2 \cdot 15,\quad 30, $

so that $ f(30)=5$. The function $ f(N)$, as a multiplicative analogue of the (additive) partition function $ p(N)$, was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Turán, and others.

Recently, Zaharescu and Zaki showed that $ f(N)$ is even a positive proportion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression $ a\operatorname {mod} m$, the set of $ N$ for which

$\displaystyle f(N) \equiv a( \operatorname {mod} m) $

possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such $ N$. For the case investigated by Zaharescu and Zaki, we show that $ f$ is odd more than 50 percent of the time (in fact, about 57 percent).

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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
Email: pppollac@illinois.edu, twonth@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11254-7
PII: S 0002-9939(2012)11254-7
Received by editor(s): May 4, 2011
Published electronically: March 15, 2012
Additional Notes: The author is supported by NSF award DMS-0802970.
Communicated by: Ken Ono
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.