A number $n$ is considered interesting if 2018 divides $d(n)$, the number of positive divisors of $n$. We aim to determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

To solve this, we need to identify the conditions on $k$ that allow for such an arithmetic progression. We will show that $k$ must satisfy one of the following two conditions:
1. There exists a prime number $p$ such that $v_p(k) \geq 2018$.
2. There exist two distinct prime numbers $q$ and $r$ such that $v_q(k) \geq 1009$ and $v_r(k) \geq 2$.

### Proof:
1. Condition 1:
If $k$ has a prime factor $p$ with $v_p(k) \geq 2018$, then consider the arithmetic progression $\{ p^{2017}(p-1 + n \frac{k}{p^{2017}}) \}_{n \in \mathbb{Z}^+}$. Here, $p \mid \frac{k}{p^{2017}}$ and $p \nmid (p-1)$. Each term in this progression will have a number of divisors divisible by 2018, making all terms interesting.

2. Condition 2:
If $k$ has two distinct prime factors $q$ and $r$ with $v_q(k) \geq 1009$ and $v_r(k) \geq 2$, then consider the arithmetic progression $\{ q^{1008}r(k-1 + n \frac{k}{q^{1008}r}) \}_{n \in \mathbb{Z}^+}$. Here, $q, r \mid \frac{k}{q^{1008}r}$ and $q, r \nmid (k-1)$. Each term in this progression will also have a number of divisors divisible by 2018, making all terms interesting.

### Inductive Step:
We use induction to show that if an infinite arithmetic progression with common difference $m$ exists, then $m$ must satisfy one of the two conditions above.

- Base Case:
For $m = 1$, there is no such arithmetic progression.

- Inductive Hypothesis:
Assume the statement is true for all $m \leq k$.

- Inductive Step:
Suppose there exists an arithmetic progression $\{ a + nd \}_{n \in \mathbb{Z}^+}$ with common difference $d = m$ satisfying the condition. If $m$ does not satisfy either condition, we derive a contradiction by considering the prime factorization of $m$ and using properties of divisors and prime numbers.

Thus, the positive integers $k$ that allow for an infinite arithmetic progression of interesting numbers must satisfy one of the two conditions stated above.

The answer is: \boxed{\text{All } k \text{ such that } v_p(k) \geq 2018 \text{ for some prime } p \text{ or } v_q(k) \geq 1009 \text{ and } v_r(k) \geq 2 \text{ for some distinct primes } q \text{ and } r.}