Divisor Equation

First, the inequality $d(n) \le 2\sqrt{n}$ implies that $\frac{n}{d(n)} \ge \frac{\sqrt{n}}2,$ so $n \le 4036^2.$ (For a proof of the inequality, see my solution to Divisor Problem (2), or just think about pairing off divisors which multiply to $n.$ )

Now $n = 2018 d(n),$ so the prime $1009$ must divide $n.$ If $1009^2 | n,$ then $1009 | d(n),$ which implies that $n$ has a prime divisor with exponent at least $1008.$ This would make $n$ far bigger than $4036^2,$ so the conclusion is that $n = 1009a$ where $a$ is coprime to $1009.$ This implies $2018 = \frac{n}{d(n)} = \frac{1009a}{d(1009a)} = \frac{1009}2 \frac{a}{d(a)},$ so $\frac{a}{d(a)} = 4.$

By the inequality from the beginning, we need only search for such $a$ up to $a = 8^2,$ and a quick search finds that there is exactly one solution, $a=36.$ Hence there is only one value of $n,$ $n = 1009 \cdot 36 = \fbox{36324}.$