14. (15 points) Find the sum of all natural numbers less than 2001 that are coprime with 2001.



保留源文本的换行和格式，直接输出翻译结果。
【Analysis】Numbers that are not coprime with 2001 must appear in pairs, and the sum of each pair is 2001. Therefore, we only need to find the number of natural numbers less than 2001 that are coprime with 2001.

【Solution】If $(a, 2001)=1$, then $(2001-a, 2001)=1$, meaning that natural numbers less than 2001 and coprime with 2001 appear in pairs. And $a+2001-a-2001$, so the sum of natural numbers less than 2001 and coprime with 2001 $=\frac{1}{2} \times$ the number of natural numbers less than 2001 and coprime with 2001 $\times 2001$. $2001=3 \times 23 \times 29$.

The number of numbers not greater than 2001 that have a common divisor of 3 with 2001 $=\left[\frac{2001}{3}\right]=667$; the number of numbers not greater than 2001 that have a common divisor of 23 with 2001 $=\left[\frac{2001}{23}\right]=87$; the number of numbers not greater than 2001 that have a common divisor of 29 with 2001 $=\left[\frac{2001}{29}\right]=69$; the number of numbers not greater than 2001 that have a common divisor of $3 \times 29$ with 2001 $=\left[\frac{2001}{3 \times 29}\right]=23$;

The number of numbers not greater than 2001 that have a common divisor of $3 \times 23$ with 2001 $=\left[\frac{2001}{3 \times 23}\right]=29$; the number of numbers not greater than 2001 that have a common divisor of $23 \times 29$ with 2001 $=\left[\frac{2001}{23 \times 29}\right]=3$; the number of numbers not greater than 2001 that have a common divisor of $3 \times 23 \times 29$ with 2001 $=\left[\frac{2001}{3 \times 23 \times 29}\right]=1$. The number of numbers less than 2001 and coprime with 2001 $=2001-67-87-69+23+29+3-1=1323$. The sum of all natural numbers less than 2001 and coprime with 2001 $=\frac{2001 \times 1232}{2}=1232616$.