Find all sequence $a_1, a_2, \ldots$ of positive integers such that
\[
	\left[\frac{a_m}{a_n}\right] = \left[\frac{m}{n}\right]
\]

for any positive integers $m$, $n$.
However, for a real number $r$, the maximum that does not exceed $r$ is denoted by $[r]$.
1. We start by analyzing the given condition:
   \[
   \left[\frac{a_m}{a_n}\right] = \left[\frac{m}{n}\right]
   \]
   for any positive integers \( m \) and \( n \). Here, \([r]\) denotes the greatest integer less than or equal to \( r \).

2. We claim that the only such sequences are \( a_i = ki \) for some fixed positive integer \( k \). To verify this, we first observe that if \( a_i = ki \), then:
   \[
   \frac{a_m}{a_n} = \frac{km}{kn} = \frac{m}{n}
   \]
   and thus:
   \[
   \left[\frac{a_m}{a_n}\right] = \left[\frac{m}{n}\right]
   \]
   which satisfies the given condition.

3. Next, we need to show that these are the only sequences that satisfy the condition. Fix any positive integer \( n \) and choose a positive integer \( N \). We have:
   \[
   0 \leq \left|\frac{a_N}{a_n} - \frac{N}{n}\right| < 1
   \]
   This inequality follows from the fact that the floor function \([x]\) is the greatest integer less than or equal to \( x \).

4. Multiplying the above inequality by \( \frac{a_n}{N} \), we get:
   \[
   0 \leq \left|\frac{a_N}{N} - \frac{a_n}{n}\right| < \frac{a_n}{N}
   \]

5. Taking the limit as \( N \to \infty \), the term \( \frac{a_n}{N} \) approaches 0. Therefore, we have:
   \[
   \lim_{N \to \infty} \left|\frac{a_N}{N} - \frac{a_n}{n}\right| = 0
   \]
   which implies:
   \[
   \lim_{N \to \infty} \frac{a_N}{N} = \frac{a_n}{n}
   \]

6. Since this holds for any fixed positive integer \( n \), it follows that:
   \[
   \frac{a_n}{n} = \frac{a_1}{1}
   \]
   for all positive integers \( n \). Let \( k = a_1 \). Then:
   \[
   a_n = kn
   \]
   for all positive integers \( n \).

7. Therefore, the only sequences that satisfy the given condition are of the form \( a_i = ki \) for some fixed positive integer \( k \).

\(\blacksquare\)

The final answer is \( \boxed{ a_i = ki } \) for some fixed positive integer \( k \).