How many different number are there in sequence $⌊n/1\rfloor$, $⌊n/2⌋, ⌊n/3⌋, ..., ⌊n/n]$? $1 \leq n \leq 10^{12}$ and n is an integer. Through some documents on the internet, I vaguely know that the answer is $\sqrt{n}$ or maybe $2\sqrt{(n)}$. Can anyone help me to prove this? My English is not very good, sorry if it makes you confused.

The answer is indeed between $\sqrt{n}$ and $2 \sqrt{n}$ (plus-minus some constants, you can figure them out if needed).
To understand this, look at $\sqrt{n}$.

*

*Clearly, there are at most $\sqrt{n}$ values which are less than $\lfloor \sqrt{n} \rfloor = \lfloor \frac{n}{\sqrt{n}} \rfloor \ge \lfloor \frac{n}{\lceil \sqrt{n} \rceil} \rfloor$. I.e., for $k \ge \lceil \sqrt{n} \rceil$, at most $\lfloor \sqrt n \rfloor$ values of $\lfloor \frac nk \rfloor$ are distinct.

*On the other hand, there are at most $\lfloor \sqrt{n} \rfloor$ distinct values $\lfloor \frac n1 \rfloor, \lfloor \frac n2 \rfloor, \ldots, \lfloor \frac{n}{\lfloor \sqrt n \rfloor} \rfloor$. Moreover, all these values are distinct, since for all $i \le \sqrt{n}$:
$$\frac{n}{i-1} - \frac{n}{i}
= \frac{n (i - (i - 1))}{i(i - 1)}
\ge \frac{n}{\sqrt n(\sqrt n - 1)}
\ge 1$$
