I began with problem which looked simple in the beginning but became increasingly complex as I dug deeper. 


**Main questions**: Find the number of solutions $s(n)$ of the equation
$$
n = \frac{k\_1}{1} + \frac{k\_2}{2} + \ldots + \frac{k\_n}{n}
$$
where $k\_i \ge 0$ is a non-negative integer. This is my main questions. After tying different approaches, the one that I found most optimistic is as follows. But soon even this turned out to be devil (as we shall see why). 


Let $l\_n$ be the LCM of the first $n$ natural numbers We know that $\log l\_n =\psi(n)$. Multiplying both sides by $l\_n$ we obtain
$$
n l\_n = \frac{k\_1 l\_n}{1} + \frac{k\_2 l\_n}{2} + \ldots + \frac{k\_n l\_n}{n}
$$


Each term on the RHS is a positive integer thus our question is equivalent to finding the number of partitions of $nl\_n$ in which each part satisfy some criteria.


**Criteria 1**: How small can a part be? Assume that there is a solution with $k\_n = 1$ then the smallest term in the above sum will be the $n$-th term which is $l\_n / n$. Hence each term in our partition is $\ge l\_n/n$.


**Criteria 2**: How many prime factors can each part contain? If my calculation is correct then for $n \ge 2, 2 \le r \le n$, the minimum number of prime factors that $l\_n /r$ can contain is $\pi(n)-1$. With these two selection criterion we have:


**$s(n) \le $ No. of partitions of $n l\_n$ into at most $n$ parts such that each part is greater than $l\_n / n$ and has at least $\pi(n) - 1$ different prime factors.**


May be we can narrow down further by adding sharper selection criterions but I thought it was already complicated enough for the time being. The asymptotics of the number of partitions of $n$ into $k$ parts $p(n,k)$ is well known, but I have not found in literature any asymptotics for the number of partitions of $n$ into $k$ parts such that each part is at least $m$, let alone the case when each part has a certain minimum number of prime factors. I am looking for any suggestions, reference materials that would help in these intermediate questions that would ultimately help in answering the main question.



A simple looking problem in partitions that became increasingly complex
I've got the following counts (which agrees with Brendan's):


1: 1


2: 3


3: 10


4: 55


5: 196


6: 2730


7: 10032


8: 108999


9: 973258


10: 20780331


11: 79309308


12: 2614200602


13: 10073335754


14: 288845706742


15: 11805287917646


16: 254331289285523


