Answer: 697

Solution:

Write $z = a + 164i.$ Clearing the denominator gives $z = 4i(z + n),$ that is, $$\begin{aligned} a + 164i &= 4i\,(a + n + 164i) \\ &= -656 + 4(a + n)i. \end{aligned}$$

Real parts give $a = -656,$ and imaginary parts give $164 = 4(a + n),$ so $a + n = 41$ and $n = 41 + 656 = 697.$