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poj3292 Semi-prime H-numbers
题意:
Description
This problem is based on an exercise of David Hilbert, who pedagogically suggested that one study the theory of 4n+1 numbers. Here, we do only a bit of that.
An H-number is a positive number which is one more than a multiple of four: 1, 5, 9, 13, 17, 21,... are the H-numbers. For this problem we pretend that these are the only numbers. The H-numbers are closed under multiplication.
As with regular integers, we partition the H-numbers into units, H-primes, and H-composites. 1 is the only unit. An H-number h is H-prime if it is not the unit, and is the product of two H-numbers in only one way: 1 × h. The rest of the numbers are H-composite.
For examples, the first few H-composites are: 5 × 5 = 25, 5 × 9 = 45, 5 × 13 = 65, 9 × 9 = 81, 5 × 17 = 85.
Your task is to count the number of H-semi-primes. An H-semi-prime is an H-number which is the product of exactly two H-primes. The two H-primes may be equal or different. In the example above, all five numbers are H-semi-primes. 125 = 5 × 5 × 5 is not an H-semi-prime, because it‘s the product of three H-primes.
Input
Each line of input contains an H-number ≤ 1,000,001. The last line of input contains 0 and this line should not be processed.
Output
For each inputted H-number h, print a line stating h and the number of H-semi-primes between 1 and h inclusive, separated by one space in the format shown in the sample.
Sample Input
21 85 789 0
Sample Output
21 0 85 5 789 62
思路:
先打H-素数表,再打只有两个素因子的表,最后二分。
实现:
1 #include <iostream> 2 #include <cstdio> 3 #include <vector> 4 #include <algorithm> 5 #include <cstring> 6 using namespace std; 7 8 const int MAXN = 1000001; 9 bool is_prime[MAXN + 5]; 10 int x; 11 vector<int> res; 12 void init() 13 { 14 vector<int> prime; 15 memset(is_prime, 1, sizeof(is_prime)); 16 for (int i = 5; i <= MAXN; i += 4) 17 { 18 if (is_prime[i]) 19 { 20 prime.push_back(i); 21 for (int j = 5; i * j <= MAXN; j += 4) 22 { 23 is_prime[i * j] = false; 24 } 25 } 26 } 27 for (int i = 25; i <= MAXN; i += 4) 28 { 29 if (is_prime[i]) 30 continue; 31 int tmp = i; 32 int cnt = 0; 33 for (int j = 0; prime[j] * prime[j] <= tmp; j++) 34 { 35 if (cnt > 2) 36 break; 37 while (tmp % prime[j] == 0) 38 { 39 cnt++; 40 tmp /= prime[j]; 41 } 42 } 43 if (tmp != 1) 44 cnt++; 45 if (cnt <= 2) 46 res.push_back(i); 47 } 48 } 49 50 int main() 51 { 52 init(); 53 while (scanf("%d", &x), x) 54 { 55 printf("%d %d\n", x, upper_bound(res.begin(), res.end(), x) - res.begin()); 56 } 57 return 0; 58 }
poj3292 Semi-prime H-numbers