ON PRODUCT PARTITIONS OF INTEGERS

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers greater-than-or-equal-to 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer d greater-than-or-equal-to 1 let d(i) = d1/i if d is an i(th) power, and = 1, otherwise, and let dBAR = PI-i = 1 infinity d(i). Using a suitable generating function for p*(n) we prove that PI-d/n d(p*(n/d))BAR = n(p*)(n).