| Let f(n) denote the number of factorizations of the natural number n > 1 into factors larger than 1 where the order of the factors does not count, and let f(1) = 1. The essentially different factorizations of n are called the multiplicative partitions of n.In 1983, Hughes and Shallit [4] proved thatand conjecturedIn 1986 and a year later, Mattics and Dodd [5], and independently Chen [2] proved thatf(n) ≤ n. In 1987, Dodd and Mattics [3] proved thatBut f(n) may have more good estimate for many integral numbers. In this paper, we estimate f(n) when P2(n) > 3, where P2(n) is the smallest prime divisor of n.Firstly, we proof the following four lemmas. Lemma 1. If n = pβ, p > 3, and β≥1, thenwhere p is a prime.Lemma 2. If n>1, thenwhere P1(n) is the largest prime divisor of n. Lemma 3. If P2(n) > 3 and w(n)≥ 2, then Lemma 4. If P2(n) > 3, n ≠175 and n≤ exp((15000)1/3), then where P2(n) is the smallest prime divisor of n. Secondly, we proof Theorem. If P2{n) > 3 and n ≠175, thenwhere P2(n) is the smallest prime factor of n. |
