素数の剰余類別集計
(number-theory/prime-residue.hpp)

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• Last update: 2026-07-25 02:01:37+09:00
• Include: #include "number-theory/prime-residue.hpp"

素数の剰余類別集計

$N$ 以下の素数の個数と総和を $M$ を法とする剰余類ごとに求める．$N\geq 0$，$M\geq 1$ とする．

• PrimeResidue::count(N, M)：(xs, count) を返す．count[i][k] は $p\leq xs[i]$ かつ $p\equiv k\pmod M$ となる素数 $p$ の個数である．
• PrimeResidue::sum<T>(N, M)：(xs, sum) を返す．sum[i][k] は $p\leq xs[i]$ かつ $p\equiv k\pmod M$ となる素数 $p$ の総和である．T の既定値は __int128_t．

xs は集合 ${\lfloor N/i\rfloor:1\leq i\leq N}$ の元を昇順に並べた列である．

T は整数から構築でき，加減算と乗算ができる型とする．

アルゴリズム

基底 $e_0,e_1,\dots,e_{M-1}$ に積 $e_a e_b=e_{ab\bmod M}$ を定める．count では完全乗法的関数 $e_{n\bmod M}$，sum では $ne_{n\bmod M}$ に Lucy DP を適用する．

いずれも時間計算量は $O\left(MN^{3/4}/\log N\right)$，空間計算量は $O(M\sqrt N)$．

Depends on

• math/util.hpp
• Lucy DP (number-theory/lucy-dp.hpp)
• 素数篩 (number-theory/prime-sieve.hpp)

Verified with

• verify/number-theory/UNIT_prime_residue.test.cpp

Code

#pragma once

#include "number-theory/lucy-dp.hpp"

namespace PrimeResidue {
using ll = long long;

namespace internal {
template <class T>
T triangular(ll n) {
  if (n % 2 == 0) return T(n / 2) * T(n + 1);
  return T(n) * T((n + 1) / 2);
}

template <class T>
struct Value {
  vector<T> value;
  int point = -1;

  Value() = default;
  explicit Value(int m) : value(m) {}

  Value operator-(const Value& rhs) const {
    int m = value.size();
    Value ret(m);
    for (int k = 0; k < m; k++) ret.value[k] = value[k] - rhs.value[k];
    return ret;
  }

  Value& operator-=(const Value& rhs) {
    int m = value.size();
    for (int k = 0; k < m; k++) value[k] -= rhs.value[k];
    point = -1;
    return *this;
  }

  Value operator*(const Value& rhs) const {
    assert(point == -1 && rhs.point != -1);
    int m = value.size(), b = rhs.point;
    Value ret(m);
    for (int a = 0; a < m; a++) {
      int k = (ll)a * b % m;
      ret.value[k] += value[a] * rhs.value[b];
    }
    return ret;
  }
};

template <class T, class F, class G>
pair<vector<ll>, vector<vector<T>>> run(ll n, int m, F point_coefficient, G prefix_coefficient) {
  if (n == 0) return {};

  function<Value<T>(ll)> point_value = [&](ll x) {
    Value<T> v(m);
    int k = x % m;
    v.value[k] = point_coefficient(x);
    v.point = k;
    return v;
  };
  function<Value<T>(ll)> prefix_sum = [&](ll x) {
    Value<T> v(m);
    for (int k = 0; k < m; k++) v.value[k] = prefix_coefficient(x, k);
    return v;
  };

  auto [xs, values] = LucyDP<Value<T>>(n, point_value, prefix_sum);
  vector<vector<T>> result;
  result.reserve(values.size());
  for (auto& value : values) result.push_back(move(value.value));
  return {move(xs), move(result)};
}
};  // namespace internal

pair<vector<ll>, vector<vector<ll>>> count(ll n, int m) {
  assert(n >= 0 && m >= 1);
  return internal::run<ll>(n, m, [](ll) { return 1LL; }, [&](ll x, int k) {
    ll rem = x % m;
    return x / m + (k > 0 && k <= rem);
  });
}

template <class T = __int128_t>
pair<vector<ll>, vector<vector<T>>> sum(ll n, int m) {
  assert(n >= 0 && m >= 1);
  return internal::run<T>(n, m, [](ll x) { return T(x); }, [&](ll x, int k) {
    ll count = x / m + (k > 0 && k <= x % m);
    if (count == 0) return T(0);
    ll first = k == 0 ? m : k;
    return T(count) * T(first) + T(m) * internal::triangular<T>(count - 1);
  });
}
};  // namespace PrimeResidue

/**
 * @brief 素数の剰余類別集計
 * @docs docs/number-theory/prime-residue.md
 */

#line 2 "number-theory/prime-residue.hpp"

#line 2 "number-theory/lucy-dp.hpp"

#line 2 "math/util.hpp"

namespace Math {
template <class T>
T safe_mod(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  a %= b;
  return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
  if (n <= 0) return 0;
  long long x = sqrt(n);
  while ((__int128)(x + 1) * (x + 1) <= n) x++;
  while ((__int128)x * x > n) x--;
  return x;
}
long long floor_root(long long n, int k) {
  assert(n >= 0);
  if (n == 0) return 0;
  assert(k >= 1);
  if (k == 1) return n;
  if (k > 64) return 1;
  long long x = round(pow((long double)n, 1.0L / k));
  auto check = [&](long long a) {
    if (a <= 0) return true;
    __int128_t p = 1;
    for (int i = 0; i < k; ++i)
      if ((p *= a) > n) return false;
    return true;
  };
  while (check(x + 1)) x++;
  while (!check(x)) x--;
  return x;
}
unsigned long long floor_root_unsigned(unsigned long long n, int k) {
  assert(k >= 1);
  if (n <= 1 || k == 1) return n;
  if (k >= 64) return 1;
  int bits = (64 + k - 1) / k;
  unsigned long long ok = 1, ng = min(n, 1ULL << bits);
  auto check = [&](unsigned long long a) {
    __uint128_t p = 1;
    for (int i = 0; i < k; i++) {
      p *= a;
      if (p > n) return false;
    }
    return true;
  };
  while (ok + 1 < ng) {
    unsigned long long mid = ok + (ng - ok) / 2;
    (check(mid) ? ok : ng) = mid;
  }
  return ok;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0  -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
  T a0 = a, b0 = b;
  bool sgn_a = a < 0, sgn_b = b < 0;
  if (sgn_a) a = -a;
  if (sgn_b) b = -b;
  if (b == 0) {
    x = sgn_a ? -1 : 1;
    y = 0;
    return a;
  }
  T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
  while (b != 0) {
    T q = a / b, r = a - b * q;
    x00 -= q * x01;
    x10 -= q * x11;
    swap(x00, x01);
    swap(x10, x11);
    a = b, b = r;
  }
  x = x00, y = x10;
  if (sgn_a) x = -x;
  if (sgn_b) y = -y;
  if (b0 != 0) {
    a0 /= a, b0 /= a;
    if (b0 < 0) a0 = -a0, b0 = -b0;
    T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
    x -= b0 * q;
    y += a0 * q;
  }
  return a;
}
constexpr long long inv_mod(long long x, long long m) {
  x %= m;
  if (x < 0) x += m;
  long long a = m, b = x;
  long long y0 = 0, y1 = 1;
  while (b > 0) {
    long long q = a / b;
    swap(a -= q * b, b);
    swap(y0 -= q * y1, y1);
  }
  if (y0 < 0) y0 += m / a;
  return y0;
}
long long pow_mod(long long x, long long n, long long m) {
  if (m == 1) return 0;
  x = (x % m + m) % m;
  long long y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
  if (m == 1) return 0;
  unsigned int _m = (unsigned int)(m);
  unsigned long long r = 1;
  unsigned long long y = x % m;
  if (y >= m) y += m;
  while (n) {
    if (n & 1) r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
constexpr bool is_prime_constexpr(int n) {
  if (n <= 1) return false;
  if (n == 2 || n == 7 || n == 61) return true;
  if (n % 2 == 0) return false;
  long long d = n - 1;
  while (d % 2 == 0) d /= 2;
  constexpr long long bases[3] = {2, 7, 61};
  for (long long a : bases) {
    long long t = d;
    long long y = pow_mod_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1) {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0) {
      return false;
    }
  }
  return true;
}
template <int n>
constexpr bool is_prime = is_prime_constexpr(n);
};  // namespace Math
#line 2 "number-theory/prime-sieve.hpp"

namespace PrimeSieve {
using ll = long long;

vector<int> lpf(int n) {
  assert(n >= 0);
  vector<int> ret(n + 1);
  for (size_t i = 0; i < ret.size(); i++) ret[i] = (int)i;
  for (int p = 2; (ll)p * p <= n; p++) {
    if (ret[p] != p) continue;
    for (ll x = (ll)p * p;; x += p) {
      if (ret[x] == x) ret[x] = p;
      if (n - x < p) break;
    }
  }
  return ret;
}

vector<int> table(int n) {
  assert(n >= 0);
  vector<bool> composite(n + 1, false);
  for (int p = 2; (ll)p * p <= n; p += (p & 1) + 1) {
    if (composite[p]) continue;
    for (ll x = (ll)p * p;; x += p) {
      composite[x] = true;
      if (n - x < p) break;
    }
  }
  vector<int> ps;
  for (int p = 2; p <= n;) {
    if (!composite[p]) ps.push_back(p);
    int step = (p & 1) + 1;
    if (n - p < step) break;
    p += step;
  }
  return ps;
}

vector<vector<pair<ll, int>>> factorize(int n) {
  assert(n >= 0);
  vector<vector<pair<ll, int>>> factors(n + 1);
  auto lp = lpf(n);
  for (int x = 2; x <= n;) {
    int y = x;
    while (y > 1) {
      int p = lp[y], e = 0;
      while (y % p == 0) y /= p, e++;
      factors[x].emplace_back(p, e);
    }
    if (x == n) break;
    x++;
  }
  return factors;
}
};  // namespace PrimeSieve

/**
 * @brief 素数篩
 * @docs docs/number-theory/prime-sieve.md
 */
#line 5 "number-theory/lucy-dp.hpp"

template <class T>
pair<vector<long long>, vector<T>> LucyDP(long long n, function<T(ll)> point_value, function<T(ll)> prefix_sum) {
  using ll = long long;
  assert(n >= 1);
  const ll sq = Math::isqrt(n);

  vector<ll> qs(sq * 2 - (n / sq == sq));
  for (int i = 0; i < sq; i++) qs[i] = i + 1;
  for (int i = 0; i < sq; i++) qs[qs.size() - 1 - i] = n / (i + 1);
  vector<T> s(qs.size());
  auto v1 = point_value(1);
  for (int i = 0; i < (int)qs.size(); i++) s[i] = prefix_sum(qs[i]) - v1;

  auto ps = PrimeSieve::table(sq);
  for (ll p : ps) {
    auto v = point_value(p);
    for (int i = (int)qs.size() - 1; i >= 0; i--) {
      ll q = qs[i];
      if (p * p > q) break;
      ll x = q / p;
      int j = x <= sq ? x - 1 : (int)s.size() - n / x;
      s[i] -= (s[j] - s[p - 2]) * v;
    }
  }

  return {qs, s};
}

/**
 * @brief Lucy DP
 * @docs docs/number-theory/lucy-dp.md
 */
#line 4 "number-theory/prime-residue.hpp"

namespace PrimeResidue {
using ll = long long;

namespace internal {
template <class T>
T triangular(ll n) {
  if (n % 2 == 0) return T(n / 2) * T(n + 1);
  return T(n) * T((n + 1) / 2);
}

template <class T>
struct Value {
  vector<T> value;
  int point = -1;

  Value() = default;
  explicit Value(int m) : value(m) {}

  Value operator-(const Value& rhs) const {
    int m = value.size();
    Value ret(m);
    for (int k = 0; k < m; k++) ret.value[k] = value[k] - rhs.value[k];
    return ret;
  }

  Value& operator-=(const Value& rhs) {
    int m = value.size();
    for (int k = 0; k < m; k++) value[k] -= rhs.value[k];
    point = -1;
    return *this;
  }

  Value operator*(const Value& rhs) const {
    assert(point == -1 && rhs.point != -1);
    int m = value.size(), b = rhs.point;
    Value ret(m);
    for (int a = 0; a < m; a++) {
      int k = (ll)a * b % m;
      ret.value[k] += value[a] * rhs.value[b];
    }
    return ret;
  }
};

template <class T, class F, class G>
pair<vector<ll>, vector<vector<T>>> run(ll n, int m, F point_coefficient, G prefix_coefficient) {
  if (n == 0) return {};

  function<Value<T>(ll)> point_value = [&](ll x) {
    Value<T> v(m);
    int k = x % m;
    v.value[k] = point_coefficient(x);
    v.point = k;
    return v;
  };
  function<Value<T>(ll)> prefix_sum = [&](ll x) {
    Value<T> v(m);
    for (int k = 0; k < m; k++) v.value[k] = prefix_coefficient(x, k);
    return v;
  };

  auto [xs, values] = LucyDP<Value<T>>(n, point_value, prefix_sum);
  vector<vector<T>> result;
  result.reserve(values.size());
  for (auto& value : values) result.push_back(move(value.value));
  return {move(xs), move(result)};
}
};  // namespace internal

pair<vector<ll>, vector<vector<ll>>> count(ll n, int m) {
  assert(n >= 0 && m >= 1);
  return internal::run<ll>(n, m, [](ll) { return 1LL; }, [&](ll x, int k) {
    ll rem = x % m;
    return x / m + (k > 0 && k <= rem);
  });
}

template <class T = __int128_t>
pair<vector<ll>, vector<vector<T>>> sum(ll n, int m) {
  assert(n >= 0 && m >= 1);
  return internal::run<T>(n, m, [](ll x) { return T(x); }, [&](ll x, int k) {
    ll count = x / m + (k > 0 && k <= x % m);
    if (count == 0) return T(0);
    ll first = k == 0 ? m : k;
    return T(count) * T(first) + T(m) * internal::triangular<T>(count - 1);
  });
}
};  // namespace PrimeResidue

/**
 * @brief 素数の剰余類別集計
 * @docs docs/number-theory/prime-residue.md
 */

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