HDU 5901 - Count Primes
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Problem Description
Easy question! Calculate how many primes between [1...n]!
Input
Each line contain one integer n(1 <= n <= 1e11).Process to end of file.
Output
For each case, output the number of primes in interval [1...n]
Sample Input
2
3
10
Sample Output
1
2
二、传说中的π函数
Easy question! Calculate how many primes between [1...n]!
Input
Each line contain one integer n(1 <= n <= 1e11).Process to end of file.
Output
For each case, output the number of primes in interval [1...n]
Sample Input
2
3
10
Sample Output
1
2
4
题意:给出一个数,求出 1 到这个数之间总共有多少个素数。
题目无比简单,数据范围无比恶心。后来得知这是模板题,在这提供两个模板。
一、快速求范围内素数
#include <cstdio>#include <algorithm>using namespace std;long long f[340000 + 5], g[340000 + 5];long long n;void Init(){ long long m; for (m = 1; m*m <= n; ++m) f[m] = n/m - 1; for (long long i = 1; i <= m; ++i) g[i] = i - 1; for (long long i = 2; i <= m; ++i) { if (g[i] == g[i-1]) continue; for (long long j = 1; j <= min(m-1, n/i/i); ++j) { if (i * j < m) f[j] -= (f[i*j] - g[i-1]); else f[j] -= (g[n/i/j] - g[i-1]); } for (long long j = m; j >= i*i; --j) g[j] -= (g[j/i] - g[i-1]); }}int main(){ while (scanf("%lld", &n) != EOF) { Init(); printf("%lld\n", f[1]); } return 0;}
二、传说中的π函数
<span style="color:#000000;">#include<cstdio>#include<cmath>using namespace std;#define LL long longconst int N = 5e6 + 2;bool np[N];int prime[N], pi[N];int getprime(){ int cnt = 0; np[0] = np[1] = true; pi[0] = pi[1] = 0; for(int i = 2; i < N; ++i) { if(!np[i]) prime[++cnt] = i; pi[i] = cnt; for(int j = 1; j <= cnt && i * prime[j] < N; ++j) { np[i * prime[j]] = true; if(i % prime[j] == 0) break; } } return cnt;}const int M = 7;const int PM = 2 * 3 * 5 * 7 * 11 * 13 * 17;int phi[PM + 1][M + 1], sz[M + 1];void init(){ getprime(); sz[0] = 1; for(int i = 0; i <= PM; ++i) phi[i][0] = i; for(int i = 1; i <= M; ++i) { sz[i] = prime[i] * sz[i - 1]; for(int j = 1; j <= PM; ++j) phi[j][i] = phi[j][i - 1] - phi[j / prime[i]][i - 1]; }}int sqrt2(LL x){ LL r = (LL)sqrt(x - 0.1); while(r * r <= x) ++r; return int(r - 1);}int sqrt3(LL x){ LL r = (LL)cbrt(x - 0.1); while(r * r * r <= x) ++r; return int(r - 1);}LL getphi(LL x, int s){ if(s == 0) return x; if(s <= M) return phi[x % sz[s]][s] + (x / sz[s]) * phi[sz[s]][s]; if(x <= prime[s]*prime[s]) return pi[x] - s + 1; if(x <= prime[s]*prime[s]*prime[s] && x < N) { int s2x = pi[sqrt2(x)]; LL ans = pi[x] - (s2x + s - 2) * (s2x - s + 1) / 2; for(int i = s + 1; i <= s2x; ++i) ans += pi[x / prime[i]]; return ans; } return getphi(x, s - 1) - getphi(x / prime[s], s - 1);}LL getpi(LL x){ if(x < N) return pi[x]; LL ans = getphi(x, pi[sqrt3(x)]) + pi[sqrt3(x)] - 1; for(int i = pi[sqrt3(x)] + 1, ed = pi[sqrt2(x)]; i <= ed; ++i) ans -= getpi(x / prime[i]) - i + 1; return ans;}LL lehmer_pi(LL x){ if(x < N) return pi[x]; int a = (int)lehmer_pi(sqrt2(sqrt2(x))); int b = (int)lehmer_pi(sqrt2(x)); int c = (int)lehmer_pi(sqrt3(x)); LL sum = getphi(x, a) +(LL)(b + a - 2) * (b - a + 1) / 2; for (int i = a + 1; i <= b; i++) { LL w = x / prime[i]; sum -= lehmer_pi(w); if (i > c) continue; LL lim = lehmer_pi(sqrt2(w)); for (int j = i; j <= lim; j++) sum -= lehmer_pi(w / prime[j]) - (j - 1); } return sum;}int main(){ init(); LL n; while(~scanf("%lld",&n)) { printf("%lld\n",lehmer_pi(n)); } return 0;}</span>
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