Some results on ordered and unordered factorization of a positive integers

As a well-known enumerative problem, the number of solutions of the equation $m=m_1+\mathrm{.}\mathrm{.}\mathrm{.}+m_k$ with $m_1⩽\mathrm{.}\mathrm{.}\mathrm{.}⩽m_k$ in positive integers is $\mathrm{\Pi }(m,k)={\sum }_{i=0}^k\mathrm{\Pi }(m-k,i)$ and $\mathrm{\Pi }$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function ${\mu }_1(m,k):=$ the number of solutions of the equation $m=m_1\mathrm{.}\mathrm{.}\mathrm{.}{m}_k$ with $m_1⩽\mathrm{.}\mathrm{.}\mathrm{.}⩽m_k$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k=1,2,3,4$.