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• W2068442839 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 2 · 3 · 5, 5 · 6, 3 · 10, 2 · 15, 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 Turan, 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 amodm, the set of N for which f(N) ≡ a(modm) 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)." @default.
• W2068442839 created "2016-06-24" @default.
• W2068442839 creator A5089571181 @default.
• W2068442839 date "2012-11-01" @default.
• W2068442839 modified "2023-09-28" @default.
• W2068442839 title "On the parity of the number of multiplicative partitions and related problems" @default.
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• W2068442839 doi "https://doi.org/10.1090/s0002-9939-2012-11254-7" @default.
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