If $a = 1, d = 2014$, then $a+x \mid 2014$. As $a+x > 1$, there are 7 possibilities.
If $a = 2, d = 1007$, then $a+x \mid 1007$. As $a+x > 2$, there are 3 possibilities.
If $a = 19, d = 106$, then $a+x \mid 106$. As $a+x > 19$, there are 2 possibilities.
If $a = 38, d = 53$, then $a+x \mid 53$. As $a+x > 38$, there is 1 possibility.
Those add up to 13 possibilities.

As the number $2014 + n$ is always a divisor of $2014 \cdot (2014 + n) = 2014^2 + 2014n$, the number $2014 + n$ is a divisor of $2014n$ if and only if $2014 + n$ is a divisor of $2014^2$. The canonical representation $2014^2 = 2^2 \cdot 19^2 \cdot 53^2$ shows that the number of divisors of $2014^2$ is $(2+1) \cdot (2+1) \cdot (2+1) = 27$. The middle one of them is 2014 and 13 divisors are expressible in a form $2014 + n$ for some positive integer $n$. Thus there are 13 possibilities for $n$.