We are given a problem involving buses and stations in Mathcity, where stations are indexed by powers of 2, such as $1, 2, 4, 8, \ldots$. Each bus visits finitely many stations, and its number is the sum of the indices of these stations. The mayor wants the bus numbers to form an arithmetic progression with the first term being his favorite number and a common difference $r$.

The key is to determine for which positive integers $r$ it is always possible, regardless of the mayor's favorite number, that for any $m$ stations, there is a bus visiting all of them.

### Analyzing the Problem

The stations are indexed by powers of 2, which implies each station has a unique binary representation consisting of a single $1$ bit.

1. The arithmetic progression of bus numbers is formed with a first term $a$ (the mayor's favorite number) and common difference $r$. The sequence of bus numbers can be expressed as:
$$a, a + r, a + 2r, a + 3r, \ldots$$

2. We need to have a bus whose number equals the sum of indices of any given subset of $m$ stations. Each such subset can be represented by a binary number with $m$ bits set to 1.

3. If a bus needs to visit exactly the stations corresponding to $m$ specific powers of 2, its number would be the sum of these powers. For example, if a bus visits stations $1$ (which is $2^0$) and $4$ (which is $2^2$), the bus number would be $1 + 4 = 5$.

### Ensuring Every Subset is Covered

The bus number can be any sum of the form:
$$\sum_{i=0}^{k} b_i \cdot 2^i$$
where each $b_i$ is either 0 or 1, indicating whether the bus visits the station at $2^i$.

To ensure any combination of stations totaling $m$ stations is represented by a bus number in the sequence, we need:
- The sequence of bus numbers $a + nr$ to cover all such binary sums uniquely.

### Proving the Necessity of an Odd $r$

To have every possible sum of $m$ distinct powers of 2 be a bus number, the arithmetic progression must span all such sums modulo 2. When $r$ is odd, $r$ adds or subtracts an odd number in successive terms, ensuring coverage of both odd and even sums.

### Conclusion

If $r$ is odd, the differences will respect the binary representation needs since every number in the sequence of bus numbers (given any starting number) can represent every possible subset by covering both parity possibilities. Therefore, all odd integers $r$ allow us to satisfy the requirement of having a bus visiting any subset of $m$ stations.

Thus, the set of all such positive integers $r$ for which this is always possible is:
$$\boxed{\text{all odd } r}$$