Linear Sieve Method for Prime Numbers

Problem description:When we calculate for prime numbers with a sieve method,we delete so many numbers which is not necessary repeatly.For instance,there is a number which consists of 3x7x17x23,and we delete it when we delete the multiples of 3 as we delete the same number when we delete the multiples of 7,17,and 23.Please write a program that will not do these jobs more than once.
   
Thinking: There is a factorization theorem:every composite number could be decomposed into the multiplication of some primer numbers.Hence,the number can be decomposed in the form of  (both of p and q are prime numbers and p < q).Therefore,what we need to remove is:,,...and,i=1,2,3.....The value of p and q is the numbers which are not removed currently and in a sequence from small to large.It is easy to write the program.

 1 #include <stdio.h>
 2 #define MAX 1000
 3 #define null1 0
 4 #define NEXT(x)  x=next[x]
 5 #define REMOVE(x) {   previous[next[x]]=previous[x];   \
 6                       next[previous[x]]=next[x];       \
 7                   }
 8 
 9 #define INITIAL(n)  { unsigned long i;                    \
10                       for(i=2;i<=n;i++)                   \
11                           previous[i]=i-1,next[i]=i+1;    \
12                       previous[2]=next[n]=null1;           \
13                     }
14 
15 int main()
16 {
17     unsigned long previous[MAX+1]={0};
18     unsigned long next[MAX+1]={0};
19     unsigned long prime,fact,i,mult;
20     unsigned long n;
21     unsigned long count=0;
22     
23     scanf("%lu",&n);
24 
25     INITIAL(n); //initial the array
26 
27     for(prime=2;prime*prime<=n;NEXT(prime))
28     {
29         for(fact=prime;prime*fact<=n;NEXT(fact)) 
30         {
31             for(mult=prime*fact;mult<=n;mult*=prime) 
32                 REMOVE(mult);
33         }
34     }
35     for(i=2;i!=null1;NEXT(i))
36         printf("%lu ",i),count++;
37     printf("\nThe sum of the prime numbers is %lu\n",count);
38 }

Reference material: C语言名题精选百则技巧篇 in Chinese.