In a few programming contexts, I've come across code along these lines:
total = 0
for i from 1 to n
total := total + n / i
Where the division here is integer division. Mathematically, this boils down to evaluating
$$\sum_{i = 1}^{n}\left\lfloor\, n \over i\,\right\rfloor.$$
This is upper-bounded by $n H_{n}$ and lower-bounded by $nH_{n} - n$ using the standard inequalities for $\texttt{floor}$'s, but that upper bound likely has a large gap to the true value.
Is there a way to either get an exact value for this summation or find some simpler function it's asymptotically equivalent to?