How many different numbers can be obtained as product of first $n$ natural numbers? Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot 2^{a_2} \cdot \ldots \cdot n^{a_n}\mid a_i\geq 0, a_1+a_2+\ldots+a_n=m \}$.
How many results can be obtained?
Denote this number with $P(m,n)$ (the number of elements of the set defined above).
For example, if $m=1$, then $P(1,n)=n$. 
Similarly, we have:
$P(m,1)=1$,
$P(m,2)=m+1$.
Moreover, if $n=p$ is prime then it is not difficult to see that 
$$P(m,p)=\sum_{i=0}^{m}P(i,p-1).$$
(We define $P(0,n)$ to be equal $1$ for any $n$.)
Furhter, using the above property one can obtain the values of $P(m,n)$, for some small values of $n$, for example for $n\leq 10$ we have:
$P(m,3)=\binom{m+2}{2}$;
$P(m,4)=(m+1)^2$;
$P(m,5)=\frac{(m+1)(m+2)(2m+3)}{6}$
$P(m,6)=(m+1){{m+2}\choose{2}}$;
$P(m,7)=\frac{(m+1)(m+2)(m+3)(3m+4)}{24}$;
$P(m,8)=\frac{(m+1)^2(m+2)(m+3)}{6}$;
$P(m,9)=\frac{(m+1)^2(m+2)^2}{4}$;
$P(m,10)=\frac{(m+1)^2(m+2)(2m+3)}{6}$.
Is there a general method that would give precise value of $P(m,n)$ for any $m$ and $n$? Or at least an approximation of $P(m,n)$?
 A: I'll consider the problem when $n$ is fixed and $m$ tends to infinity.  Here the examples provided by the OP seem to suggest that $P(m,n)$ behaves like a polynomial in $m$ of degree $\pi(n)$.  I'll prove that (again thinking that $n$ is small, and $m$ is large) 
$$ 
(1+o(1)) \frac{m^{\pi(n)}}{\pi(n)!} \prod_{p\le n} \lfloor \frac{\log n}{\log p}\rfloor 
\le P(m,n) \le (1+o(1)) \frac{m^{\pi(n)}}{\pi(n)!} \prod_{p\le n} \frac{\log n}{\log p}.
$$ 
For example when $n=10$ this shows that 
$$ 
P(m,10) \le (1+o(1)) \frac{m^4}{24} \frac{(\log 10)^4}{(\log 2)(\log 3)(\log 5)(\log 7)} = (0.4911\ldots+o(1)) m^4,
$$ 
and 
$$ 
P(m,10) \ge (1+o(1)) \frac{m^4}{4},
$$ 
which are in keeping with the asymptotic $P(m,10) \sim m^4/3$ found in the question. 
Edit:  Here is a better upper bound for $P(m,n)$: 
$$ 
P(m,n) \le (1+o(1)) \frac{m^{\pi(n)}}{\pi(n)!} \prod_{p\le n} \lfloor \frac{\log n}{\log p}\rfloor (\pi(n))^{\pi(\sqrt{n})}.
$$
Combining the upper and lower bounds, we now have for fixed $n$ and $m$ large
$$ 
P(m,n) = \frac{m^{\pi(n)}}{\pi(n)!} \exp(O(\sqrt{n})).
$$
Now for the proofs.  For the original upper bound, as observed by Eric Naslund, one has $P(m,n) \le \Psi(n^m,n)$ where $\Psi(x,y)$ denotes the number of integers below $x$ all of whose prime factors lie below $y$.  Naturally $\Psi(x,y)$ has been extensively studied.  In the range we are interested in, $y$ is kept very small (less than $\log x$) and $x$ is much bigger.  Here the problem of counting smooth numbers should be thought of as a problem of counting lattice points inside a high dimensional ``tetrahedron."  This is an old idea, and for smooth numbers was worked out by Ennola.  There is a paper by Granville in Aequationes Math. which discusses such lattice point problems in many related situations: see http://www.dms.umontreal.ca/~andrew/PDF/tetrahedron.pdf (specifically, eqn (2.4) there).  Note that $\Psi(n^m,n)$ counts all lattice points in $\pi(n)$ dimensional space such that the coordinates are nonnegative and $\sum_{p\le n} a_p \log p \le m\log n$.  The number of such lattice points is asymptotically the volume of this tetrahedron, and this gives our upper bound.  
As for the lower bound, I claim that any number of the form $\prod_{p\le n} p^{a_p}$ with the property that $a_p$'s are non-negative integers and
$$   
\sum_{p\le n} a_p \Big(\lfloor \frac{\log n}{\log p} \rfloor\Big)^{-1} \le m-\pi(n),
$$ 
is counted in $P(m,n)$.  To see this, for each prime $p\le n$ put $b_p=\lfloor \log n/\log p\rfloor$ and use $[a_p/b_p]$ values of $p^{b_p}$ and one extra power of $p$ to get the exponents to add up to $a_p$; the assumed inequality guarantees that at most $m$ numbers are used in doing this. So once again we are counting the lattice points in a tetrahedron, and computing the volume of this tetrahedron we obtain the lower bound. 
Now for the improved upper bound.  Divide the primes below $n$ into the sets $P_{k}=\{n^{1/(k+1)}<p\le n^{1/k}\}$ for $1\le k\le \log n/\log 2$.  Suppose a number $N=\prod_{p\le n} p^{a_p}$ is counted in $P(m,n)$.  Then we see that for each $1\le k\le \log n/\log 2$ we must have $\sum_{p\in P_k} a_p \le km$.  The number of non-negative $a_p$ with $\sum_{p\in P_k} a_p\le km$ is $\binom{km+|P_k|}{km}=\binom{km+|P_k|}{|P_k|}$.  Thus we conclude that 
$$ 
P(m,n) \le \prod_{k\le \log n/\log 2} \binom{km+|P_k|}{|P_k|}.
$$ 
Now for $k=1$ we have 
$$ 
\binom{m+|P_1|}{|P_1|} = (1+o(1)) \frac{m^{|P_1|}}{|P_1|!} \le (1+o(1)) \frac{m^{|P_1|}}{\pi(n)!} \pi(n)^{\pi(\sqrt{n})}.
$$
And for larger $k$ simply use that 
$$ 
\binom{km+|P_k|}{|P_k|} \le (1+o(1))(km)^{|P_k|}.
$$
The new upper bound follows. 
A: I add this as a new answer as it has a somewhat different flavor from my previous response.  Moreover the bounds given there may still be useful.  Thanks also to Edgardo for a useful discussion.  
We consider the case when $n$ is fixed and $m$ is large.  First we translate the problem into one of counting lattice points in certain polytopes.  It will then follow from work of Khovanskii that for $P(m,n)$ here (for $m$ large) is given by a polynomial in $m$ of degree $\pi(n)$ and whose leading coefficient will be described below.  It may be that for $P(m,n)$ is in fact a polynomial from the start, and maybe just the Erhart polynomial attached to a convex polytope defined below (this last point now seems unlikely; see the remark at the end of the answer).  
To each natural number below $n$ associate a vector in ${\Bbb Z}^{\pi(n)}$ with non-negative coordinates corresponding to the exponents in the prime factorization of that number.  Thus when $n=5$ we have the vectors $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(2,0,0)$ and $(0,0,1)$.  Let ${\mathcal C}(n)$ denote the convex hull of these $n$ vectors.  If a number $N$ is written as a product of $m$ numbers below $n$, then the prime factorization of $N$ gives a vector which is contained in the set ${\mathcal C}(n)$ dilated by $m$.  Therefore it follows that the $P(m,n)$ is at most the number of lattice points contained in $m\cdot {\mathcal C}(n)$ which is asymptotically 
$$ 
m^{\pi(n)} \text{Vol}({\mathcal C}(n)).
$$
Further note that any vector in $(m-n)\cdot {\mathcal C}(n)$ can be expressed as a sum of at most $m$ of the $n$ vectors corresponding to $1$ to $n$.  Therefore $P(m,n)$ is at least as large as the number of lattice points contained in $(m-n) \cdot {\mathcal C}(n)$ and this also is asymptotically 
$$ 
m^{\pi(n)} \text{Vol}({\mathcal C}(n)).
$$ 
Thus we have shown that $P(m,n) \sim m^{\pi(n)} \text{Vol}({\mathcal C}(n))$.  My previous answer may be seen as giving upper and lower bounds for the volume of ${\mathcal C}(n)$ by bounding it from above and below by simplices. 
In view of the above translation, we see that the problem may be phrased generally as follows:  Given a commutative semigroup $G$ and two finite subsets $A$ and $B$ of $G$, consider all elements that are the sum of an element of $B$ and $N$ elements of $A$; call this set $B+N*A$.  Theorem 1 of Khovanskii's paper (Newton polyhedron, Hilbert polynomial, and Sums of Finite sets; translation from Russian of his paper in Functional analysis & Applications 1992) then shows that the number of elements in $B+N*A$ is a polynomial in $N$ for large $N$.  Khovanskii also considers the special situation when $G$ is ${\Bbb Z}^n$ and $A$ is a finite subset of $G$ which generates all of $G$. This is the situation of the problem at hand, and his work in Section 3 of the cited paper is along the lines of the argument I gave above.  Another interesting paper on this topic is the work of Barvinok and Woods (Short rational generating functions for lattice point problems) which led me to Khovanskii's paper.   
Edit:  See also my related question An integrality question about expressing an integer as a product of numbers below $n$ ; especially Ilya Bogdanov's excellent example which indicates that $P(m,n)$ for larger values of $n$ is perhaps not a polynomial from the start and also probably not the Erhart polynomial for ${\mathcal C}(n)$.
A: If you fix $m$, this is known as the $m$-dimensional multiplication problem. In 2010 Koukoulopoulos showed that as $n\rightarrow \infty$ $$P(m,n)=\left|\lbrace a_1\cdots a_m\ :\ a_i\leq n \text{ for all } \ i\rbrace\right|\asymp \frac{n^{m+1}}{(\log n)^{c_m}(\log\log n)^{3/2}}$$ where $$c_{m}=\int_{1}^{\frac{k}{\log(m+1)}}\log x\text{d}x=\frac{\log(m+1)+m\log\left(m\right)-m\log\log(m+1)-m}{\log(m+1)}.$$ See this answer for more details.
For $m\rightarrow \infty$, and $n$ fixed your calculations show that the order of magnitude looks like $m^{\pi(n)}$ where $\pi(n)=\sum_{p\leq n}1$ denotes the number of primes less that $n$. By considering those primes $p\leq n$, it is easy to see that we have a lower bound for $P(n,m)$ of this form. In what follows, by using Rankin's trick I will obtain the bound $$P(n,m)\leq m^{\pi(n)} e^{2n}\ \ \ \ \ \ \ \ \ \ \ (1) $$ which holds for any $m$ and $n\geq 3$. (Of course this bound is only strong when $n$ is very small compared to $m$)  From this, it follows that for $n$ fixed and $m\rightarrow \infty$ we obtain the correct order of magnitude $$P(n,m)\asymp m^{\pi(n)}.$$
Notice that every element in your set lies in ${1,\dots,n^m}$, and has no prime factor larger than $n$. Let $S(y)=\{n\in\mathbb{Z}:P(n)\leq y\}$ where $P(n)$ is the largest prime factor of $n$. Then $$P(n,m)\leq \sum_{\begin{array}{c}
k\leq n^{m}\\
k\in S(n)
\end{array}}1.$$
For any $\sigma>0$, since $\sum_{n=1,\ n\in S(y)}^\infty n^{-\sigma}=\prod_{p\leq y} \left(1-1/p^\sigma\right)^{-1}$, we have that 
$$P(n,m)\leq \sum_{\begin{array}{c}
k\leq n^{m}\\
k\in S(n)
\end{array}}\frac{n^{\sigma m}}{k^\sigma}=n^{\sigma m} \prod_{p\leq n}\left(1-\frac{1}{p^\sigma}\right)^{-1}.$$ Since $1-p^{-\sigma}\geq\frac{\sigma\log p}{2}$, it follows that 
$$\prod_{p\leq n}\left(1-p^{-\sigma}\right)^{-1}\leq2^{\pi(n)}\sigma^{-\pi(n)}\prod_{p\leq n}\frac{1}{\log p}\leq2^{\pi(n)}\sigma^{-\pi(n)}. $$
Choosing $\sigma=\pi(n)/m$, it follows that $$P(n,m)\leq m^{\pi(n)}n^{\pi(n)}2^{\pi(n)}\pi(n)^{-\pi(n)}.$$ Applying the Brun-Titchmarsh theorem, we obtain equation $(1)$ for all $m,n$ with $n\geq 3$.
