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:heavy_check_mark: nt/array_on_divisors.hpp

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#include "nt/factor.hpp"
#include "ds/hashmap.hpp"

template <typename T>
struct Array_On_Divisors {
  vc<pair<ll, int>> pf;
  vc<ll> divs;
  vc<T> dat;
  HashMap<int> MP;

  Array_On_Divisors(ll N = 1) { build(N); }
  Array_On_Divisors(vc<pair<ll, int>> pf) { build(pf); }

  void build(ll N) { build(factor(N)); }
  void build(vc<pair<ll, int>> pfs) {
    if (!pf.empty() && pf == pfs) return;
    pf = pfs;
    ll n = 1;
    for (auto&& [p, e]: pf) n *= (e + 1);
    divs.assign(n, 1);
    dat.assign(n, T{});
    int nxt = 1;
    for (auto&& [p, e]: pf) {
      int L = nxt;
      ll q = p;
      FOR(e) {
        FOR(i, L) { divs[nxt++] = divs[i] * q; }
        q *= p;
      }
    }
    MP.build(n);
    FOR(i, n) MP[divs[i]] = i;
  }

  T& operator[](ll d) { return dat[MP[d]]; }

  // f(p, k) を与える → 乗法的に拡張
  template <typename F>
  void set_multiplicative(F f) {
    dat.reserve(len(divs));
    dat = {T(1)};
    for (auto&& [p, e]: pf) {
      int n = len(divs);
      FOR(k, 1, e + 1) { FOR(i, n) dat.eb(dat[i] * f(p, k)); }
    }
  }

  void set_euler_phi() {
    dat.resize(len(divs));
    FOR(i, len(divs)) dat[i] = T(divs[i]);
    divisor_mobius();
  }

  void set_mobius() {
    set_multiplicative([&](ll p, int k) -> T {
      if (k >= 2) return T(0);
      return (k == 1 ? T(-1) : T(0));
    });
  }

  void multiplier_zeta() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j] += dat[mod * i + j + k]; }
      }
      k *= (e + 1);
    }
  }

  void multiplier_mobius() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j] -= dat[mod * i + j + k]; }
      }
      k *= (e + 1);
    }
  }

  void divisor_zeta() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j + k] += dat[mod * i + j]; }
      }
      k *= (e + 1);
    }
  }

  void divisor_mobius() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j + k] -= dat[mod * i + j]; }
      }
      k *= (e + 1);
    }
  }

  // (Ta,Tb)->T : a-b
  template <typename F>
  void divisor_mobius(F SUB) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j + k] = SUB(dat[mod * i + j + k], dat[mod * i + j]); }
      }
      k *= (e + 1);
    }
  }

  // ADD(Ta,Tb)->T : a+b
  template <typename F>
  void multiplier_zeta(F ADD) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j] = ADD(dat[mod * i + j], dat[mod * i + j + k]); }
      }
      k *= (e + 1);
    }
  }

  // SUB(Ta,Tb)->T : a-=b
  template <typename F>
  void multiplier_mobius(F SUB) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j] = SUB(dat[mod * i + j], dat[mod * i + j + k]); }
      }
      k *= (e + 1);
    }
  }

  // ADD(T&a,Tb)->void : a+=b
  template <typename F>
  void divisor_zeta(F ADD) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j + k] = ADD(dat[mod * i + j + k], dat[mod * i + j]); }
      }
      k *= (e + 1);
    }
  }

  template <typename F>
  void set(F f) {
    FOR(i, len(divs)) { dat[i] = f(divs[i]); }
  }

  // (d, fd)
  // &fd で受け取れば代入とかもできます
  template <typename F>
  void enumerate(F f) {
    FOR(i, len(divs)) { f(divs[i], dat[i]); }
  }
};
#line 2 "nt/factor.hpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "mod/mongomery_modint.hpp"

// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
  using mint = Mongomery_modint;
  inline static U1 m, r, n2;
  static constexpr int W = numeric_limits<U1>::digits;

  static void set_mod(U1 mod) {
    assert(mod & 1 && mod <= U1(1) << (W - 2));
    m = mod, n2 = -U2(m) % m, r = m;
    FOR(5) r *= 2 - m * r;
    r = -r;
    assert(r * m == U1(-1));
  }
  static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }

  U1 x;
  Mongomery_modint() : x(0) {}
  Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
  U1 val() const {
    U1 y = reduce(x);
    return y >= m ? y - m : y;
  }
  mint &operator+=(mint y) {
    x = ((x += y.x) >= m ? x - m : x);
    return *this;
  }
  mint &operator-=(mint y) {
    x -= (x >= y.x ? y.x : y.x - m);
    return *this;
  }
  mint &operator*=(mint y) {
    x = reduce(U2(x) * y.x);
    return *this;
  }
  mint operator+(mint y) const { return mint(*this) += y; }
  mint operator-(mint y) const { return mint(*this) -= y; }
  mint operator*(mint y) const { return mint(*this) *= y; }
  bool operator==(mint y) const {
    return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
  }
  bool operator!=(mint y) const { return not operator==(y); }
  mint pow(ll n) const {
    assert(n >= 0);
    mint y = 1, z = *this;
    for (; n; n >>= 1, z *= z)
      if (n & 1) y *= z;
    return y;
  }
};

template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 3 "nt/primetest.hpp"

bool primetest(const u64 x) {
  assert(x < u64(1) << 62);
  if (x == 2 or x == 3 or x == 5 or x == 7) return true;
  if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
  if (x < 121) return x > 1;
  const u64 d = (x - 1) >> lowbit(x - 1);

  using mint = Mongomery_modint_64<202311020>;

  mint::set_mod(x);
  const mint one(u64(1)), minus_one(x - 1);
  auto ok = [&](u64 a) -> bool {
    auto y = mint(a).pow(d);
    u64 t = d;
    while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1;
    if (y != minus_one && t % 2 == 0) return false;
    return true;
  };
  if (x < (u64(1) << 32)) {
    for (u64 a: {2, 7, 61})
      if (!ok(a)) return false;
  } else {
    for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
      if (!ok(a)) return false;
    }
  }
  return true;
}
#line 5 "nt/factor.hpp"

template <typename mint>
ll rho(ll n, ll c) {
  assert(n > 1);
  const mint cc(c);
  auto f = [&](mint x) { return x * x + cc; };
  mint x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1LL << (__lg(n) / 5);
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(r) y = f(y);
    for (ll k = 0; k < r && g == 1; k += m) {
      z = y;
      FOR(min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n) do {
      z = f(z);
      g = gcd((x - z).val(), n);
    } while (g == 1);
  return g;
}

ll find_prime_factor(ll n) {
  assert(n > 1);
  if (primetest(n)) return n;
  FOR(100) {
    ll m = 0;
    if (n < (1 << 30)) {
      using mint = Mongomery_modint_32<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    } else {
      using mint = Mongomery_modint_64<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    }
    if (primetest(m)) return m;
    n = m;
  }
  assert(0);
  return -1;
}

// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
  assert(n >= 1);
  vc<pair<ll, int>> pf;
  FOR(p, 2, 100) {
    if (p * p > n) break;
    if (n % p == 0) {
      ll e = 0;
      do { n /= p, e += 1; } while (n % p == 0);
      pf.eb(p, e);
    }
  }
  while (n > 1) {
    ll p = find_prime_factor(n);
    ll e = 0;
    do { n /= p, e += 1; } while (n % p == 0);
    pf.eb(p, e);
  }
  sort(all(pf));
  return pf;
}

vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
  vc<pair<ll, int>> res;
  while (n > 1) {
    int p = lpf[n];
    int e = 0;
    while (n % p == 0) {
      n /= p;
      ++e;
    }
    res.eb(p, e);
  }
  return res;
}
#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 3 "nt/array_on_divisors.hpp"

template <typename T>
struct Array_On_Divisors {
  vc<pair<ll, int>> pf;
  vc<ll> divs;
  vc<T> dat;
  HashMap<int> MP;

  Array_On_Divisors(ll N = 1) { build(N); }
  Array_On_Divisors(vc<pair<ll, int>> pf) { build(pf); }

  void build(ll N) { build(factor(N)); }
  void build(vc<pair<ll, int>> pfs) {
    if (!pf.empty() && pf == pfs) return;
    pf = pfs;
    ll n = 1;
    for (auto&& [p, e]: pf) n *= (e + 1);
    divs.assign(n, 1);
    dat.assign(n, T{});
    int nxt = 1;
    for (auto&& [p, e]: pf) {
      int L = nxt;
      ll q = p;
      FOR(e) {
        FOR(i, L) { divs[nxt++] = divs[i] * q; }
        q *= p;
      }
    }
    MP.build(n);
    FOR(i, n) MP[divs[i]] = i;
  }

  T& operator[](ll d) { return dat[MP[d]]; }

  // f(p, k) を与える → 乗法的に拡張
  template <typename F>
  void set_multiplicative(F f) {
    dat.reserve(len(divs));
    dat = {T(1)};
    for (auto&& [p, e]: pf) {
      int n = len(divs);
      FOR(k, 1, e + 1) { FOR(i, n) dat.eb(dat[i] * f(p, k)); }
    }
  }

  void set_euler_phi() {
    dat.resize(len(divs));
    FOR(i, len(divs)) dat[i] = T(divs[i]);
    divisor_mobius();
  }

  void set_mobius() {
    set_multiplicative([&](ll p, int k) -> T {
      if (k >= 2) return T(0);
      return (k == 1 ? T(-1) : T(0));
    });
  }

  void multiplier_zeta() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j] += dat[mod * i + j + k]; }
      }
      k *= (e + 1);
    }
  }

  void multiplier_mobius() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j] -= dat[mod * i + j + k]; }
      }
      k *= (e + 1);
    }
  }

  void divisor_zeta() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j + k] += dat[mod * i + j]; }
      }
      k *= (e + 1);
    }
  }

  void divisor_mobius() {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j + k] -= dat[mod * i + j]; }
      }
      k *= (e + 1);
    }
  }

  // (Ta,Tb)->T : a-b
  template <typename F>
  void divisor_mobius(F SUB) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j + k] = SUB(dat[mod * i + j + k], dat[mod * i + j]); }
      }
      k *= (e + 1);
    }
  }

  // ADD(Ta,Tb)->T : a+b
  template <typename F>
  void multiplier_zeta(F ADD) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR_R(j, mod - k) { dat[mod * i + j] = ADD(dat[mod * i + j], dat[mod * i + j + k]); }
      }
      k *= (e + 1);
    }
  }

  // SUB(Ta,Tb)->T : a-=b
  template <typename F>
  void multiplier_mobius(F SUB) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j] = SUB(dat[mod * i + j], dat[mod * i + j + k]); }
      }
      k *= (e + 1);
    }
  }

  // ADD(T&a,Tb)->void : a+=b
  template <typename F>
  void divisor_zeta(F ADD) {
    ll k = 1;
    for (auto&& [p, e]: pf) {
      ll mod = k * (e + 1);
      FOR(i, len(divs) / mod) {
        FOR(j, mod - k) { dat[mod * i + j + k] = ADD(dat[mod * i + j + k], dat[mod * i + j]); }
      }
      k *= (e + 1);
    }
  }

  template <typename F>
  void set(F f) {
    FOR(i, len(divs)) { dat[i] = f(divs[i]); }
  }

  // (d, fd)
  // &fd で受け取れば代入とかもできます
  template <typename F>
  void enumerate(F f) {
    FOR(i, len(divs)) { f(divs[i], dat[i]); }
  }
};
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