Question 164
For each positive integer, define a function
$$
f(n) = \begin{cases} 0, & \text{if} n \text{ is the square of an integer,} \\ \left\lfloor \frac{1}{\sqrt{n}} \right\rfloor, & \text{if} n \text{ is not the square of an integer.} \end{cases}
$$
(Here $[x]$ denotes the maximum integer not exceeding $x$, and
$\{x\} = x - [x]$.) Find the value of $\sum_{k=1}^{200} f(k)$.
$$
f(n) = \begin{cases} 0, & \text{if} n \text{ is the square of an integer,} \\ \left\lfloor \frac{1}{\sqrt{n}} \right\rfloor, & \text{if} n \text{ is not the square of an integer.} \end{cases}
$$
(Here $[x]$ denotes the maximum integer not exceeding $x$, and
$\{x\} = x - [x]$.) Find the value of $\sum_{k=1}^{200} f(k)$.