Tifa's CP Library

:heavy_check_mark: src/test_cpverifier/library-checker/counting_squarefrees.test.cpp

Depends on

Code

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

#include "../../code/nt/sqrfree_cnt.hpp"

int main() {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  u64 n;
  std::cin >> n;
  std::cout << tifa_libs::math::sqrfree_cnt(n) << '\n';
  return 0;
}
#line 1 "src/test_cpverifier/library-checker/counting_squarefrees.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/counting_squarefrees"

#line 1 "src/code/nt/sqrfree_cnt.hpp"



#line 1 "src/code/math/ikth_root.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/stdc++.h>

template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;

using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;

using f32 = float;
using f64 = double;
using f128 = long double;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
template <class T>
using vec = std::vector<T>;
template <class T>
using vvec = vec<vec<T>>;
template <class T>
using v3ec = vec<vvec<T>>;
template <class U, class T>
using vecp = vec<std::pair<U, T>>;
template <class U, class T>
using vvecp = vvec<std::pair<U, T>>;
template <class T>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<T>>;

template <class T, class C = std::less<T>>
using pq = std::priority_queue<T, vec<T>, C>;
template <class T>
using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using namespace std::literals;

constexpr i8 operator""_i8(unsigned long long x) { return (i8)x; }
constexpr i16 operator""_i16(unsigned long long x) { return (i16)x; }
constexpr i32 operator""_i32(unsigned long long x) { return (i32)x; }
constexpr i64 operator""_i64(unsigned long long x) { return (i64)x; }
constexpr isz operator""_iz(unsigned long long x) { return (isz)x; }

constexpr u8 operator""_u8(unsigned long long x) { return (u8)x; }
constexpr u16 operator""_u16(unsigned long long x) { return (u16)x; }
constexpr u32 operator""_u32(unsigned long long x) { return (u32)x; }
constexpr u64 operator""_u64(unsigned long long x) { return (u64)x; }
constexpr usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};


#line 5 "src/code/math/ikth_root.hpp"

namespace tifa_libs::math {

constexpr u64 ikth_root(u64 a, u64 k) {
  assert(k);
  if (a <= 1 || k == 1) return a;
  if (k >= 64) return 1;
  auto chk = [&](u128 n) {
    u128 x = 1, m = n;
    for (u64 p = k; p; p /= 2, m *= m)
      if (p & 1) x *= m;
    return x <= a;
  };
  u64 n = (u64)powl(a, 1.l / k);
  while (!chk(n)) --n;
  while (chk(n + 1)) ++n;
  return n;
}

}  // namespace tifa_libs::math


#line 1 "src/code/math/isqrt.hpp"



#line 5 "src/code/math/isqrt.hpp"

namespace tifa_libs::math {

constexpr u32 isqrt(u64 x) {
  if (!x) return 0;
  int c = i32(std::bit_width(x) - 1) / 2, sh = 31 - c;
  u32 u = [](u64 x) {
    constexpr u8 TAB[192] = {128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145, 146, 147, 148, 149, 150, 151, 151, 152, 153, 154, 155, 156, 156, 157, 158, 159, 160, 160, 161, 162, 163, 164, 164, 165, 166, 167, 167, 168, 169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 179, 179, 180, 181, 181, 182, 183, 183, 184, 185, 186, 186, 187, 188, 188, 189, 190, 190, 191, 192, 192, 193, 194, 194, 195, 196, 196, 197, 198, 198, 199, 200, 200, 201, 201, 202, 203, 203, 204, 205, 205, 206, 206, 207, 208, 208, 209, 210, 210, 211, 211, 212, 213, 213, 214, 214, 215, 216, 216, 217, 217, 218, 219, 219, 220, 220, 221, 221, 222, 223, 223, 224, 224, 225, 225, 226, 227, 227, 228, 228, 229, 229, 230, 230, 231, 232, 232, 233, 233, 234, 234, 235, 235, 236, 237, 237, 238, 238, 239, 239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 255};
    u32 u = TAB[(x >> 56) - 64];
    u = (u << 7) + (u32)(x >> 41) / u;
    return (u << 15) + (u32)((x >> 17) / u);
  }(x << 2 * sh);
  u >>= sh;
  u -= (u64)u * u > x;
  return u;
}

}  // namespace tifa_libs::math


#line 1 "src/code/nt/lsieve_func.hpp"



#line 1 "src/code/nt/lsieve.hpp"



#line 5 "src/code/nt/lsieve.hpp"

namespace tifa_libs::math {

template <class F1, class F2, class F3>
requires requires(F1 cb_prime, F2 cb_coprime, F3 cb_not_coprime, u32 p, u32 q) {
  cb_prime(p);
  cb_coprime(p, q);
  cb_not_coprime(p, q);
}
constexpr vecu lsieve(u32 n, F1 cb_prime, F2 cb_coprime, F3 cb_not_coprime) {
  vecb vis(n);
  vecu p;
  p.reserve(n <= 170 ? 16 : n / 10);
  for (u32 i = 2; i < n; ++i) {
    if (!vis[i]) {
      p.push_back(i);
      cb_prime(i);
    }
    for (u32 j : p) {
      if (i * j >= n) break;
      vis[i * j] = true;
      if (i % j) cb_coprime(i, j);
      else {
        cb_not_coprime(i, j);
        break;
      }
    }
  }
  return p;
}

}  // namespace tifa_libs::math


#line 5 "src/code/nt/lsieve_func.hpp"

namespace tifa_libs::math {

enum lsieve_state {
  ls_mpf = 1,
  ls_phi = 2,
  ls_mu = 4,
  ls_sigma = 8,
  ls_tau = 16
};

struct lsieve_func {
  vecu prime, mpf, phi;
  vec<i32> mu;
  vecu64 sigma, tau;

  template <int state>
  constexpr lsieve_func& reset_lsieve_func(u32 n) {
    if constexpr (state | ls_mpf) mpf = vecu(n), mpf[1] = 1;
    if constexpr (state | ls_phi) phi = vecu(n), phi[1] = 1;
    if constexpr (state | ls_mu) mu = vec<i32>(n), mu[1] = 1;
    if constexpr (state | ls_sigma) pw = vecu64(n), sigma = vecu64(n), sigma[1] = 1;
    if constexpr (state | ls_tau) pc = vecu(n, 1), tau = vecu64(n), tau[1] = 1;

    prime = lsieve(
        n,
        [&](u32 p) {
          if constexpr (state | ls_mpf) mpf[p] = p;
          if constexpr (state | ls_phi) phi[p] = p - 1;
          if constexpr (state | ls_mu) mu[p] = -1;
          if constexpr (state | ls_sigma) pw[p] = (u64)p * p, sigma[p] = p + 1;
          if constexpr (state | ls_tau) tau[p] = 2;
        },
        [&](u32 i, u32 j) {
          if constexpr (state | ls_mpf) mpf[i * j] = j;
          if constexpr (state | ls_phi) phi[i * j] = phi[i] * (j - 1);
          if constexpr (state | ls_mu) mu[i * j] = -mu[i];
          if constexpr (state | ls_sigma) pw[i * j] = (u64)j * j, sigma[i * j] = sigma[i] * (j + 1);
          if constexpr (state | ls_tau) tau[i * j] = tau[i] * 2;
        },
        [&](u32 i, u32 j) {
          if constexpr (state | ls_mpf) mpf[i * j] = j;
          if constexpr (state | ls_phi) phi[i * j] = phi[i] * j;
          if constexpr (state | ls_sigma) sigma[i * j] = sigma[i] * ((pw[i * j] = pw[i] * j) - 1) / (pw[i] - 1);
          if constexpr (state | ls_tau) tau[i * j] = tau[i] + tau[i] / (pc[i * j] = pc[i] + 1);
        });
    return *this;
  }

 private:
  vecu64 pw;
  vecu pc;
};

}  // namespace tifa_libs::math


#line 7 "src/code/nt/sqrfree_cnt.hpp"

namespace tifa_libs::math {

constexpr u64 sqrfree_cnt(u64 n) {
  if (n <= 3) return n;
  const u32 I = (u32)std::min(ikth_root(n, 5) * 2, ikth_root(n / 4, 3)), D = isqrt(n / I);
  auto mu = lsieve_func{}.reset_lsieve_func<ls_mu>(D + 1).mu;
  u64 ans = 0;
  for (u32 i = 1; i <= D; ++i)
    if (mu[i]) ans += (u64)mu[i] * (n / ((u64)i * i));
  for (u32 i = 1; i <= D; ++i) mu[i] += mu[i - 1];
  vec<i32> mu_large(I + 1);
  for (u32 i = I - 1; i >= 1; --i) {
    const u32 xi = isqrt(n / i), h = isqrt(xi), tlim = 2 * h - (h == xi / h);
    auto f = [xi, h, tlim](u32 i) -> u32 { return i < h ? i + 1 : u32(f64(xi) / (2 * h - (tlim & 1) - i)); };
    i64 sum = 1;
    for (u32 t = 1, l = 1; t < tlim; ++t) {
      u32 r = f(t), q = f(tlim - t - 1);
      sum -= i64(r - l) * (q <= D ? mu[q] : mu_large[i * r * r]);
      l = r;
    }
    ans += u64(mu_large[i] = (i32)sum);
  }
  return ans - u64(i64(I - 1) * mu[D]);
}

}  // namespace tifa_libs::math


#line 4 "src/test_cpverifier/library-checker/counting_squarefrees.test.cpp"

int main() {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  u64 n;
  std::cin >> n;
  std::cout << tifa_libs::math::sqrfree_cnt(n) << '\n';
  return 0;
}

Test cases

Env Name Status Elapsed Memory
g++-12 example_00 :heavy_check_mark: AC 9 ms 4 MB
g++-12 example_01 :heavy_check_mark: AC 8 ms 4 MB
g++-12 max_00 :heavy_check_mark: AC 407 ms 489 MB
g++-12 max_01 :heavy_check_mark: AC 406 ms 489 MB
g++-12 max_02 :heavy_check_mark: AC 406 ms 488 MB
g++-12 max_03 :heavy_check_mark: AC 404 ms 488 MB
g++-12 max_04 :heavy_check_mark: AC 406 ms 489 MB
g++-12 random_00 :heavy_check_mark: AC 353 ms 427 MB
g++-12 random_01 :heavy_check_mark: AC 247 ms 296 MB
g++-12 random_02 :heavy_check_mark: AC 356 ms 432 MB
g++-12 random_03 :heavy_check_mark: AC 136 ms 158 MB
g++-12 random_04 :heavy_check_mark: AC 235 ms 281 MB
g++-12 small_00 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_01 :heavy_check_mark: AC 8 ms 4 MB
g++-12 small_02 :heavy_check_mark: AC 8 ms 4 MB
g++-12 small_03 :heavy_check_mark: AC 8 ms 4 MB
g++-12 small_04 :heavy_check_mark: AC 8 ms 4 MB
g++-12 very_small_00 :heavy_check_mark: AC 8 ms 4 MB
g++-12 very_small_01 :heavy_check_mark: AC 8 ms 4 MB
g++-12 very_small_02 :heavy_check_mark: AC 8 ms 4 MB
g++-12 very_small_03 :heavy_check_mark: AC 8 ms 4 MB
g++-12 very_small_04 :heavy_check_mark: AC 8 ms 4 MB
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