結果

問題 No.8123 Calculated N !
ユーザー 👑 hos.lyric
提出日時 2025-04-01 21:55:59
言語 C++14
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 133 ms / 2,000 ms
コード長 6,376 bytes
コンパイル時間 1,494 ms
コンパイル使用メモリ 117,536 KB
実行使用メモリ 7,720 KB
最終ジャッジ日時 2025-04-01 21:56:03
合計ジャッジ時間 3,512 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge4
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 6
other AC * 16
権限があれば一括ダウンロードができます

ソースコード

diff # 

#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <chrono>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

using namespace std;

using Int = long long;

template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
#define COLOR(s) ("\x1b[" s "m")

////////////////////////////////////////////////////////////////////////////////
template <unsigned M_> struct ModInt {
  static constexpr unsigned M = M_;
  unsigned x;
  constexpr ModInt() : x(0U) {}
  constexpr ModInt(unsigned x_) : x(x_ % M) {}
  constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
  constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
  constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
  ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
  ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
  ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
  ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
  ModInt pow(long long e) const {
    if (e < 0) return inv().pow(-e);
    ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
  }
  ModInt inv() const {
    unsigned a = M, b = x; int y = 0, z = 1;
    for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
    assert(a == 1U); return ModInt(y);
  }
  ModInt operator+() const { return *this; }
  ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
  ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
  ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
  ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
  ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
  template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
  template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
  template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
  template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
  explicit operator bool() const { return x; }
  bool operator==(const ModInt &a) const { return (x == a.x); }
  bool operator!=(const ModInt &a) const { return (x != a.x); }
  friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
////////////////////////////////////////////////////////////////////////////////

constexpr unsigned MO = 1000000007;
using Mint = ModInt<MO>;


inline long long divide(long long a, int b) {
  return a / b;
}
inline long long divide(long long a, long long b) {
  return a / b;
}

Int N;
int sqrtN;
vector<int> small0;
vector<Int> large0;
inline Int pi0(Int n) { return (n <= sqrtN) ? small0[n] : large0[divide(N, n)]; }
int primesLen;
vector<int> primes;

void build() {
  sqrtN = sqrt(static_cast<double>(N));
  small0.assign(sqrtN + 1, 0);
  for (int k = 1; k <= sqrtN; ++k) {
    small0[k] = k;
  }
  large0.assign(sqrtN + 1, 0);
  for (int l = 1; l <= sqrtN; ++l) {
    const Int n = divide(N, l);
    large0[l] = n;
  }
  // \sum_{p<=n} f(p)  for n = floor(N/*)
  //   f: completely multiplicative
  //   g(p, n) := \sum[1<=x<=n] [x: prime || lpf(x) > p] f(x)
  //   g(1, n) = \sum[1<=x<=n] f(x)  (need to be computed quickly)
  //   for p: prime:
  //     g(p, n) = | g(p-1, n)                                     (n <  p^2)
  //               | g(p-1, n) - f(p) (g(p-1, n/p) - g(p-1, p-1))  (n >= p^2)
  //   O(N^(3/4) / log(N)) time, O(N^(1/2)) space
  for (int p = 2; p <= sqrtN; ++p) if (small0[p - 1] < small0[p]) {
    const int g0 = small0[p - 1];
    const int snp = sqrtN / p;
    const int psnp = p * snp;
    const long long np = divide(N, p);
    const int limL = min<long long>(divide(np, p), sqrtN);
    for (int l = 1; l <= snp; ++l) {
      const int pl = p * l;
      large0[l] -=      (large0[pl] - g0);
    }
    for (int l = snp + 1; l <= limL; ++l) {
      const int npl = divide(np, l);
      large0[l] -=      (small0[npl] - g0);
    }
    if (snp >= p) {
      for (int r = sqrtN - psnp; r >= 0; --r) {
        small0[psnp + r] -=      (small0[snp] - g0);
      }
      for (int k = snp; --k >= p; ) for (int r = p; --r >= 0; ) {
        small0[p * k + r] -=      (small0[k] - g0);
      }
    }
  }
  for (int k = 1; k <= sqrtN; ++k) {
    small0[k] -= 1;
  }
  for (int l = 1; l <= sqrtN; ++l) {
    large0[l] -= 1;
  }
  primesLen = small0[sqrtN];
  primes.resize(primesLen);
  for (int p = 2; p <= sqrtN; ++p) if (small0[p - 1] < small0[p]) {
    primes[small0[p - 1]] = p;
  }
}



int main() {
  for (; ~scanf("%lld", &N); ) {
    build();
cerr<<"pi(N) = "<<pi0(N)<<endl;
    
    Mint ans = 1;
    for (int k = 1; k <= sqrtN; ++k) {
      const Int res = pi0(N / k) - pi0(N / (k + 1));
      ans *= Mint(k + 1).pow(res);
    }
    const Int lim = N / (sqrtN + 1);
    for (const int p : primes) {
      Int e = 0;
      for (Int n = N; n /= p; ) e += n;
// if(N<=10)cerr<<p<<": "<<e<<endl;
      if (p > lim) ans /= (N / p + 1);
      ans *= (e + 1);
    }
    printf("%u\n", ans.x);
  }
  return 0;
}
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