Math/tests/prime_pi.test.cpp

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• Last update: 2022-12-26 15:37:34+08:00
• Problem: https://judge.yosupo.jp/problem/counting_primes

Depends on

• Math/Prime/PrimePi.h

Code

#define PROBLEM "https://judge.yosupo.jp/problem/counting_primes"

#include <bits/stdc++.h>
using namespace std;

#include "../Prime/PrimePi.h"

int32_t main() {
    ios::sync_with_stdio(0); cin.tie(0);
    long long n; cin >> n;
    cout << prime_pi(n) << endl;
    return 0;
}

#line 1 "Math/tests/prime_pi.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/counting_primes"

#include <bits/stdc++.h>
using namespace std;

#line 1 "Math/Prime/PrimePi.h"
// prime_pi(n) = number of primes <= n
//
// Copied from https://judge.yosupo.jp/submission/61551
//
// Tested:
// - https://judge.yosupo.jp/problem/counting_primes
// - https://www.spoj.com/problems/NTHPRIME/ (binary search + prime pi)
//
// Notes:
// - There's simpler O(N^0.75) code for prime_pi and prime_sum
//   Write up: https://www.facebook.com/code.cung.rr/posts/pfbid0eAyeQynkVN9evzy7Bnwx52zLeN7EHDE6H9Uur6KTSK2MTiyxJwCV71Wvujqz75vgl
//   Implementation:
//   - prime_sum: https://www.spoj.com/status/SUMPRIM1,mr_invincible/
//   - prime_pi: https://www.spoj.com/problems/DIVCNT2/
//   This can be modified to implement more complicated things such as
//   https://www.spoj.com/problems/DIVFACT4/

// prime_pi {{{
using ll = long long;
int isqrt(ll n) {
    return sqrtl(n);
}
ll prime_pi(const ll N) {
    if (N <= 1) return 0;
    if (N == 2) return 1;
    const int v = isqrt(N);
    int s = (v + 1) / 2;
    vector<int> smalls(s);
    for (int i = 1; i < s; i++) smalls[i] = i;
    vector<int> roughs(s);
    for (int i = 0; i < s; i++) roughs[i] = 2 * i + 1;
    vector<ll> larges(s);
    for (int i = 0; i < s; i++) larges[i] = (N / (2 * i + 1) - 1) / 2;
    vector<bool> skip(v + 1);
    const auto divide = [](ll n, ll d) -> int { return (double)n / d;};
    const auto half = [](int n) -> int { return (n - 1) >> 1;};
    int pc = 0;
    for (int p = 3; p <= v; p += 2) if (!skip[p]) {
        int q = p * p;
        if ((ll)q * q > N) break;
        skip[p] = true;
        for (int i = q; i <= v; i += 2 * p) skip[i] = true;
        int ns = 0;
        for (int k = 0; k < s; k++) {
            int i = roughs[k];
            if (skip[i]) continue;
            ll d = (ll)i * p;
            larges[ns] = larges[k] - (d <= v ? larges[smalls[d >> 1] - pc] : smalls[half(divide(N, d))]) + pc;
            roughs[ns++] = i;
        }
        s = ns;
        for (int i = half(v), j = ((v / p) - 1) | 1; j >= p; j -= 2) {
            int c = smalls[j >> 1] - pc;
            for (int e = (j * p) >> 1; i >= e; i--) smalls[i] -= c;
        }
        pc++;
    }
    larges[0] += (ll)(s + 2 * (pc - 1)) * (s - 1) / 2;
    for (int k = 1; k < s; k++) larges[0] -= larges[k];
    for (int l = 1; l < s; l++) {
        ll q = roughs[l];
        ll M = N / q;
        int e = smalls[half(M / q)] - pc;
        if (e < l + 1) break;
        ll t = 0;
        for (int k = l + 1; k <= e; k++)
            t += smalls[half(divide(M, roughs[k]))];
        larges[0] += t - (ll)(e - l) * (pc + l - 1);
    }
    return larges[0] + 1;
}
// }}}
#line 7 "Math/tests/prime_pi.test.cpp"

int32_t main() {
    ios::sync_with_stdio(0); cin.tie(0);
    long long n; cin >> n;
    cout << prime_pi(n) << endl;
    return 0;
}

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