2. (2006 National High School Mathematics League Zhejiang Province Preliminary Competition) Let $f(n)$ be the sum of the squares of the digits of the positive integer $n$ (in decimal notation), for example, $f(123)=1^{2}+2^{2}+3^{2}=14$. Denote $f_{1}(n)=f(n), f_{k+1}(n)=$ $f\left(f_{k}(n)\right), k=1,2,3, \cdots$, then the value of $f_{2005}(2006)$ is
A. 20
B. 4
C. 42
D. 145
2. D From special cases: $f(2006)=40, f_{2}(2006)=f(40)=16$,

Let $f(2006)=40$ be denoted as $2006 \rightarrow 40$, then
$$
2006 \rightarrow 40 \rightarrow 16 \rightarrow 37 \rightarrow 58 \rightarrow 89 \rightarrow 145 \rightarrow 42 \rightarrow 20 \rightarrow 4 \rightarrow 16 \rightarrow \cdots
$$

Starting from 16, $f(n)$ is a periodic sequence with a period of 8. Therefore,
$f_{\text {Yams }}(2006)=f_{\text {Zool }}(16)=f_{1 \cdot 2 \operatorname{nos}}(16)=f_{1}(16)=145$. Hence, the answer is D.
In fact, for any positive integer $n$ (in decimal), the sum of the squares of its digits, according to the problem's requirements, will eventually either form a periodic sequence with a period of 8 or result in $1,1,1 \cdots$.