Thaddeus Function

Algebra Level 4

Let
$f(n) = \frac {1}{ \sqrt{1} + \sqrt{2}} + \frac {1}{\sqrt{2} + \sqrt{3}} + \frac {1}{\sqrt{3} + \sqrt{4}} \ldots \frac{1}{\sqrt{n}+\sqrt{n+1}}.$

For how many positive integers $n$ , in the range $1 \leq n \leq 1000$ , is $f(n)$ an integer?

This problem is posed by Thaddeus A.