some results on ordered and unordered factorization of a positive integer

A well-known enumerative problem is to count the number of ways a positive integer $n$ can be factorised as $n=n_1times n_2timescdotstimes n_{k}$, where $n_1geqslant n_2 geqslant cdots geqslant n_{k} >1$. in this paper, we give some recursive formulas for the number of ordered/unordered factorizations of a positiveinteger $n$ such that each factor is at least $ell$. in particular, by using elementary techniques, we give an explicit formula in cases where $k=2,3,4$.