How many sequences a of length $N$ consisting of positive integers satisfy $a_1 \times a_2 \times … \times a_n = M$?

You are given positive integers $$N$$ and $$M$$ .

How many sequences a of length N consisting of positive integers satisfy $$a_1 \times a_2 \times ... \times a_n = M$$ ?

Here , two sequences $$a'$$ and $$a''$$ are considered different when there exists some $$i$$ such that $$ai' != ai''$$ .

For example $$N = 2$$ and $$M = 6$$, the answer is $$4$$.

$$\{a_1 , a_2 \} = \{ (1,6) (2,3) (3,2) (6,1) \}$$

In your example, the 2 could appearas $$(1,2)$$ or $$(2,1)$$, and the 3 could be $$(1,3)$$ or $$(3,1)$$. Multiply them together, for example $$(1,2)\times(1,3)=(1\times1,2\times3)=(1,6)$$.
Now how many ways can $$p^k$$ be placed in $$N$$ spots? This is a good use of stars and bars.