What's is the name of this function? A function, $f:\mathbb{N}\to\mathbb{N}$, is defined in the following way,
\begin{align}
f(n)=\#\{m\mid m\leq n, \text{ and there does not exists an integer }m' \in (m,n]\text{ such that } m|m'\}.
\end{align}
I want to know the name of this function, and I am also curious about properties of this function.
Thank everyone for helping.

As hinted at by Rob in the comments, your function can more succinctly be written as $f(n) = \lceil n/2 \rceil$. 


*

*Every number $a \leq \frac{1}{2}n$ will not be counted, because $a|2a$ and $2a \leq n$.

*Every number $a > \frac{1}{2}n$ will be counted because every number $a$ divides will be greater than $n$: $$n < 2a < 3a < 4a < \dotsb \,.$$


So then for each $n$ how many positive integers $a$ are there such that $\frac{1}{2}n < a \leq n$? Play around for a bit, and you'll see that there are $\lceil n/2 \rceil$ of them, where $\lceil x \rceil$ is the ceiling of $x$ and denotes the smallest integer greater than or equal to $x$. 
