Given $n$ and $B_n= \lceil H_n \rceil$, where the latter is the $n$th harmonic number $\sum^n_{i=1} 1/i$, for most $n$ it is easy to pack the first $n$ terms of the harmonic series into $B_n$ many unit bins. An interesting question is for which $n$ one cannot perform such a packing. My guess is there are no such $n$, and I would like to know of references to this problem.

However, even more I would like to know about exact packing. There is no exact packing for $n=4$ or $5$, but (because $1= 1/2 + 1/3 + 1/6$) there are exact packings for other $n \leq 10$. Here a packing is exact if all but (at most) one bin is exactly filled to capacity.

What is the next $n \gt 10$ for which there is an exact packing?

Are there any references to this specific problem?

Gerhard "Not Going On A Trip" Paseman, 2017.01.25.