On the parity of the number of multiplicative partitions and related problems. (English) Zbl 1330.11061
Summary: 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
\[
2\cdot 3\cdot 5,\quad 5\cdot 6,\quad 3\cdot 10, \quad 2 \cdot 15,\quad 30,
\]
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, A. Zaharescu and M. Zaki [Acta Arith. 145, No. 3, 221–232 (2010; Zbl 1248.11078)] 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 \(a\bmod m\), the set of \(N\) for which \(f(N) \equiv a\pmod m\) 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 (loc. cit.), we show that \(f\) is odd more than 50 percent of the time (in fact, about 57 percent).
MSC:
| 11N64 | Other results on the distribution of values or the characterization of arithmetic functions |
| 11B73 | Bell and Stirling numbers |
| 11P83 | Partitions; congruences and congruential restrictions |
Citations:
Zbl 1248.11078Online Encyclopedia of Integer Sequences:
The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).Number of perfect partitions of n.
Parity of the multiplicative partition function A001055.
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