Find the number of $m$ for which holds that $\lfloor{\sqrt m}\rfloor$ divides $m$

Question

For a fixed $n$ find the number of $m$'s for which $\lfloor{\sqrt m}\rfloor \mid m$, for all natural numbers $m\le n$. I have been trying to write it as $m=\lfloor{\sqrt m}\rfloor \cdot k$, where $k$ is a natural number. Putting it in the ussual floor inequalities doesn't help me much.

Answer 1

In many cases, writing out some of the actual solutions may help you to see the pattern. I think this is one of those cases.

Consider the numbers for which $\lfloor \sqrt{m} \rfloor = 4$, say. These are the numbers 16 through 24. There are exactly three numbers in this range divisible by $4$: $16, 20, 24$.

Next, consider the numbers for which $\lfloor \sqrt{m} \rfloor = 5$. These are the numbers 25 through 35. Again, there are exactly three numbers in this range divisible by $5$: $25, 30, 35$.

You might observe, then, that for any $m$, there are always three integers evenly divisible by $m$ in the range $[m^2, (m+1)^2-1]$: $m^2$, $m^2+m$, and $m^2+2m$.

Therefore, let $p = \lfloor \sqrt{n} \rfloor$. The answer is $3p$ for $n = p^2 + 2p$; it is $3p-1$ for $n$ in the range $[p^2+p, p^2+2p-1]$; and it is $3p-2$ for $n$ in the range $[p^2, p^2+p-1]$.

Answer 2

First note that $m$ can take all the perfect square values. Assume that $m$ is npt a perfect square. So there exist a unique $t$ such that

$t^2<m<(t+1)^2$

assume that

$m=t^2+k$ where $k\in{\mathbb{N}}$ and $k<2t+1$

so $\lfloor\sqrt{m}\rfloor=t$

now $\lfloor\sqrt{m}\rfloor|m$

so $t|t^2+k$

so $t|k$.

But $k<2t+1$

$k=t$ or $k=2t$

Hence there are two such possible $m$ between two perfect squares i.e. $t^2+t$ and $t^2+2t$ where $t^2<m<(t+1)^2$.

Now if $n=a^2$ then obviously there are $a$ perfect square $m$ and $2(a-1)$ non perfect square.

Hence there are $3a-2$ such $m$ for $n=a^2$

now $n$ is not a perfect square. We can uniquely find $a$ such that

$a^2<n<(a+1)^2$ or in other words $\lfloor{\sqrt{n}}\rfloor=a$

case-$1$

$a^2<n<a^2+a$

In this case again there would be $3a-2$ possible cases by the same logic.

case-$2$

$a^2+a\le{n<a^2+2a}$

In this case, by simple counting, there would be $3a-1$ possibilities

case-$3$

$a^2+2a\le{n}<(a+1)^2$

In this case there would be $3a$ possible $m$.