In algebraic number theory we come across following formula:


$n= e\_1f\_1+\cdots+e\_rf\_r$


where all $e\_i$ and $f\_i$ are positive integers. I am sure writing a positive integer n as above must be studied. 


For example :
1=1.1


2=1.1+1.1 = 1.2 = 2.1


3=1.1+1.1+1.1 = 1.1+1.2 = 1.1 +2.1 = 1.3 = 3.1


I am looking for a formula and generating function for this. Any help regarding this will be highly appreciated. Thank you.



Reference : Partition of integer
1) If we do not care on the order of summands (it looks to be the case). 


For each ordered pair $(e,f)$ take a sum $1+x^{ef}+x^{2ef}+\dots$, generating function is a product of these guys over all $(e,f)$. This may be further rewritten in different ways. At first, we may fix $e$, then we have $\prod\_f (1-x^{ef})^{-1}=P(x^{e})$, where $P(t)=\prod (1-t^f)^{-1}=\sum p(n)t^n$ is the Euler generating function for partitions. So, the answer is $\prod\_e P(x^e)$. It is appropriate to get asymptotics. We may instead fix $ef=n$, then we get $\prod\_n (1-x^n)^{-\tau(n)}$, where $\tau(n)$ is a number of divisors function. 


Of course, both functions are written on the OEIS page from the comment by Brendan McKay. 


2) If we care on the order of summands.


For $r=1$ the number of representations is number of divisors of $n$, thus generating function equals $h(x)=\sum\_{k\geqslant 1} x^k/(1-x^k)$. For arbitrary $r$ generating function if $h^r$, thus if $r$ varies, we finally get
$$
h+h^2+\dots=\frac{h}{1-h}.
$$


