Counting number of factorizations of a natural number

Similarly, there are 9 ways of factorizing 36: 36 = 2.2.3.3 = 2.2.9 = 2.3.6 = 3.3.4 = 2.18 = 3.12 = 4.9 = 6.6 = 36. Our problem is to find this number for any natural number n. Since we are not distinguishing between 2 · 9 and 9 · 2, such a factorization is called unordered. A partition of a natural number n is a representation of n as the sum of any number of positive integral parts, where the order of the parts is irrelevant. The number of such partitions of n is known as the partition function and is denoted by p(n). Likewise the function p∗(n) denotes the number of ways of expressing n as a product of positive integers greater than 1, the order of the factors in the product being irrelevant. For convenience, p∗(1) is assumed to be 1. Clearly p∗(n) is the number of unordered factorizations of n. In 1983, Hughes and Shalit [6] obtained a bound for p∗(n), namely, p∗(n) 6 2n √ 2 which was then improved to p∗(n) 6 n by Mattics and Dodd [7] in 1986. By this time Canfield, Erdös and Pomerance [2] modified a result of Oppenheim regarding the maximal order of p∗(n). They obtained another bound for p∗(n) and described an algorithm for it. An average estimate for p∗(n) was given by Oppenheim [8] which was also proved independently by Szekeres and Turan [9]. Finally, in 1991, Harris and Subbarao [5] came with a generating function and a recursion formula for p∗(n). One may consider [1] and [3] for some problems