OMM 2016 Problem 2

A pair of positive integers $m, n$ is called guerrera, if there exists positive integers $a, b, c, d$ such that $m=ab$ , $n=cd$ and $a+b=c+d$ . For example the pair $8, 9$ is guerrera cause $8= 4 \cdot 2$ , $9= 3 \cdot 3$ and $4+2=3+3$ . We paint the positive integers if the following order:

We start painting the numbers $3$ and $5$ . If a positive integer $x$ is not painted and a positive $y$ is painted such that the pair $x, y$ is guerrera, we paint $x$ .

Find all positive integers $x$ that can be painted.