test/verify/yosupo-counting-primes.test.cpp

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• Last update: 2024-07-28 19:56:12+09:00
• Problem: https://judge.yosupo.jp/problem/counting_primes

Depends on

• Kth Root Integer (math/number-theory/kth-root-integer.hpp)
• Prime Count(素数の個数) (math/number-theory/prime-count.hpp)
• Prime Table(素数テーブル) (math/number-theory/prime-table.hpp)
• template/template.hpp

Code

// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/counting_primes

#include "../../template/template.hpp"

#include "../../math/number-theory/prime-count.hpp"

int main() {
  int64_t n;
  cin >> n;
  PrimeCount<> pc;
  cout << pc.pi(n) << "\n";
}

#line 1 "test/verify/yosupo-counting-primes.test.cpp"
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/counting_primes

#line 1 "template/template.hpp"
#include <bits/stdc++.h>
#if __has_include(<atcoder/all>)
#include <atcoder/all>
#endif

using namespace std;

using int64 = long long;

const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;

struct IoSetup {
  IoSetup() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(10);
    cerr << fixed << setprecision(10);
  }
} iosetup;

template <typename T1, typename T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
  os << p.first << " " << p.second;
  return os;
}

template <typename T1, typename T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  for (int i = 0; i < (int)v.size(); i++) {
    os << v[i] << (i + 1 != v.size() ? " " : "");
  }
  return os;
}

template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (T &in : v) is >> in;
  return is;
}

template <typename T1, typename T2>
inline bool chmax(T1 &a, T2 b) {
  return a < b && (a = b, true);
}

template <typename T1, typename T2>
inline bool chmin(T1 &a, T2 b) {
  return a > b && (a = b, true);
}

template <typename T = int64>
vector<T> make_v(size_t a) {
  return vector<T>(a);
}

template <typename T, typename... Ts>
auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}

template <typename T, typename V>
typename enable_if<is_class<T>::value == 0>::type fill_v(T &t, const V &v) {
  t = v;
}

template <typename T, typename V>
typename enable_if<is_class<T>::value != 0>::type fill_v(T &t, const V &v) {
  for (auto &e : t) fill_v(e, v);
}

template <typename F>
struct FixPoint : F {
  explicit FixPoint(F &&f) : F(std::forward<F>(f)) {}

  template <typename... Args>
  decltype(auto) operator()(Args &&...args) const {
    return F::operator()(*this, std::forward<Args>(args)...);
  }
};

template <typename F>
inline decltype(auto) MFP(F &&f) {
  return FixPoint<F>{std::forward<F>(f)};
}
#line 4 "test/verify/yosupo-counting-primes.test.cpp"

#line 1 "math/number-theory/kth-root-integer.hpp"
uint64_t kth_root_integer(uint64_t a, int k) {
  if (k == 1) return a;
  auto check = [&](uint32_t x) {
    uint64_t mul = 1;
    for (int j = 0; j < k; j++) {
      if (__builtin_mul_overflow(mul, x, &mul)) return false;
    }
    return mul <= a;
  };
  uint64_t ret = 0;
  for (int i = 31; i >= 0; i--) {
    if (check(ret | (1u << i))) ret |= 1u << i;
  }
  return ret;
}
#line 1 "math/number-theory/prime-table.hpp"
/**
 * @brief Prime Table(素数テーブル)
 *
 */
vector<bool> prime_table(int n) {
  vector<bool> prime(n + 1, true);
  if (n >= 0) prime[0] = false;
  if (n >= 1) prime[1] = false;
  for (int i = 2; i * i <= n; i++) {
    if (!prime[i]) continue;
    for (int j = i * i; j <= n; j += i) {
      prime[j] = false;
    }
  }
  return prime;
}
#line 3 "math/number-theory/prime-count.hpp"

/**
 * @brief Prime Count(素数の個数)
 */
template <int64_t LIM = 100000000000LL>
struct PrimeCount {
 private:
  int64_t sq;
  vector<bool> prime;
  vector<int64_t> prime_sum, primes;

  int64_t p2(int64_t x, int64_t y) {
    if (x < 4) return 0;
    int64_t a = pi(y);
    int64_t b = pi(kth_root_integer(x, 2));
    if (a >= b) return 0;
    int64_t sum = (a - 2) * (a + 1) / 2 - (b - 2) * (b + 1) / 2;
    for (int64_t i = a; i < b; i++) sum += pi(x / primes[i]);
    return sum;
  }

  int64_t phi(int64_t m, int64_t n) {
    if (m < 1) return 0;
    if (n > m) return 1;
    if (n < 1) return m;
    if (m <= primes[n - 1] * primes[n - 1]) return pi(m) - n + 1;
    if (m <= primes[n - 1] * primes[n - 1] * primes[n - 1] && m <= sq) {
      int64_t sx = pi(kth_root_integer(m, 2));
      int64_t ans = pi(m) - (sx + n - 2) * (sx - n + 1) / 2;
      for (int64_t i = n; i < sx; ++i) ans += pi(m / primes[i]);
      return ans;
    }
    return phi(m, n - 1) - phi(m / primes[n - 1], n - 1);
  }

 public:
  PrimeCount() : sq(kth_root_integer(LIM, 2)), prime_sum(sq + 1) {
    prime = prime_table(sq);
    for (int i = 1; i <= sq; i++) prime_sum[i] = prime_sum[i - 1] + prime[i];
    primes.reserve(prime_sum[sq]);
    for (int i = 1; i <= sq; i++)
      if (prime[i]) primes.push_back(i);
  }

  int64_t pi(int64_t n) {
    if (n <= sq) return prime_sum[n];
    int64_t m = kth_root_integer(n, 3);
    int64_t a = pi(m);
    return phi(n, a) + a - 1 - p2(n, m);
  }
};
#line 6 "test/verify/yosupo-counting-primes.test.cpp"

int main() {
  int64_t n;
  cin >> n;
  PrimeCount<> pc;
  cout << pc.pi(n) << "\n";
}

Test cases

Env	Name	Status	Elapsed	Memory
g++	Grothendieck_prime_00	AC	8 ms	6 MB
g++	boundaryA_00	AC	215 ms	6 MB
g++	boundaryA_01	AC	126 ms	6 MB
g++	boundaryA_02	AC	132 ms	6 MB
g++	boundaryB_00	AC	214 ms	6 MB
g++	boundaryB_01	AC	126 ms	6 MB
g++	boundaryB_02	AC	131 ms	6 MB
g++	example_00	AC	8 ms	6 MB
g++	example_01	AC	7 ms	6 MB
g++	max_00	AC	219 ms	6 MB
g++	max_01	AC	218 ms	6 MB
g++	max_02	AC	219 ms	6 MB
g++	max_03	AC	219 ms	6 MB
g++	max_04	AC	219 ms	6 MB
g++	random_00	AC	216 ms	6 MB
g++	random_01	AC	127 ms	6 MB
g++	random_02	AC	132 ms	6 MB
g++	random_03	AC	101 ms	6 MB
g++	random_04	AC	52 ms	6 MB
g++	small_00	AC	8 ms	6 MB
g++	small_01	AC	7 ms	6 MB
g++	small_02	AC	7 ms	6 MB
g++	small_03	AC	7 ms	6 MB
g++	small_04	AC	7 ms	6 MB
g++	small_prime_00	AC	7 ms	6 MB
g++	small_prime_01	AC	7 ms	6 MB
g++	very_small_00	AC	7 ms	6 MB
g++	very_small_01	AC	7 ms	6 MB
g++	very_small_02	AC	7 ms	6 MB
g++	very_small_03	AC	7 ms	6 MB
g++	very_small_04	AC	7 ms	6 MB
clang++	Grothendieck_prime_00	AC	7 ms	6 MB
clang++	boundaryA_00	AC	222 ms	6 MB
clang++	boundaryA_01	AC	131 ms	6 MB
clang++	boundaryA_02	AC	136 ms	6 MB
clang++	boundaryB_00	AC	223 ms	6 MB
clang++	boundaryB_01	AC	135 ms	6 MB
clang++	boundaryB_02	AC	136 ms	6 MB
clang++	example_00	AC	7 ms	6 MB
clang++	example_01	AC	7 ms	6 MB
clang++	max_00	AC	228 ms	6 MB
clang++	max_01	AC	228 ms	6 MB
clang++	max_02	AC	228 ms	6 MB
clang++	max_03	AC	228 ms	6 MB
clang++	max_04	AC	228 ms	6 MB
clang++	random_00	AC	223 ms	6 MB
clang++	random_01	AC	131 ms	6 MB
clang++	random_02	AC	137 ms	6 MB
clang++	random_03	AC	106 ms	6 MB
clang++	random_04	AC	54 ms	6 MB
clang++	small_00	AC	7 ms	6 MB
clang++	small_01	AC	7 ms	6 MB
clang++	small_02	AC	7 ms	6 MB
clang++	small_03	AC	7 ms	6 MB
clang++	small_04	AC	7 ms	6 MB
clang++	small_prime_00	AC	7 ms	6 MB
clang++	small_prime_01	AC	7 ms	6 MB
clang++	very_small_00	AC	7 ms	6 MB
clang++	very_small_01	AC	7 ms	6 MB
clang++	very_small_02	AC	7 ms	6 MB
clang++	very_small_03	AC	7 ms	6 MB
clang++	very_small_04	AC	7 ms	6 MB

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