#1185843 (C++23) No.3691 Calculate Mu Sum

• 問題: No.3691 Calculate Mu Sum
• ユーザー: miscalc
• 提出日時: 2026-09-11 05:19:13
• 言語: C++23 (gcc 15.3.0 + boost 1.92.0 + ACL)
• 結果: AC
• 実行時間: 4 ms / 2,000 ms
• コード長: 46,616 bytes
• コンパイル時間: 4,322 ms
• コンパイル使用メモリ: 377,504 KB
• 実行使用メモリ: 6,400 KB
• 最終ジャッジ日時: 2026-09-11 05:19:19
• 合計ジャッジ時間: 6,071 ms
• ジャッジサーバーID （参考情報）: judge2_1 / judge1_0

ファイルパターン	結果
sample	AC * 3
other	AC * 11

ソースコード

#define SINGLE_TESTCASE 

#define FAST_IO 

#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11

// https://github.com/miscalculation53/library/tree/wip/template/template_all.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_all_but_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_types.hpp

#include <bits/stdc++.h>
using namespace std;

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;

#define vc vector
template <class T>
using vvc = vc<vc<T>>;

using vstr = vc<string>;

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (int i = neg; i < (int)s.size(); i++)
    res = 10 * res + s[i] - '0';
  if (neg)
    return -i128(res - 1) - 1;
  return i128(res);
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  u128 ux;
  if (x < 0)
    ux = u128(-(x + 1)) + 1, sign = "-";
  else
    ux = x;
  while (ux > 0)
  {
    res += '0' + ux % 10;
    ux /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  is >> s;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}

#define cauto const auto
// https://github.com/miscalculation53/library/tree/wip/template/template_rep.hpp

#define overload4(_1,_2,_3,_4,name,...) name
#define rep1(i,n) for (ll i = 0, nnnnn = ll(n); i < nnnnn; i++)
#define rep2(i,l,r) for (ll i = ll(l), rrrrr = ll(r); i < rrrrr; i++)
#define rep3(i,l,r,d) for (ll i = ll(l), rrrrr = ll(r), ddddd = ll(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i,n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i,l,r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi3(i,l,r,d) for (int i = int(l), rrrrr = int(r), ddddd = int(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
// https://github.com/miscalculation53/library/tree/wip/template/template_math.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/is_integral_ext.hpp

template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;

template <class T>
constexpr bool is_signed_ext = is_signed_v<T> || is_same_v<T, i128>;

template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;
// https://github.com/miscalculation53/library/tree/wip/utils/default_infty.hpp

namespace default_infty_detail
{

  template <class T>
  inline constexpr bool unsupported = false;
}

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b);

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;
  if (k == 2)
  {
    const T aa = T(a);
    T x = T(sqrtl((long double)a));
    while (x > aa / x)
      x--;
    while (x < numeric_limits<T>::max())
    {
      const T y = x + 1;
      if (y > aa / y)
        break;
      x = y;
    }
    return x;
  }

  auto isok = [&](T x) -> bool
  {
    if (x == 0)
      return true;
    T res = 1, k2 = k;
    while (true)
    {
      if (k2 & 1)
      {
        if (res > T(a) / x)
          return false;
        res *= x;
      }
      k2 >>= 1;
      if (k2 == 0)
        break;
      if (x > T(a) / x)
        return false;
      x *= x;
    }
    return res <= T(a);
  };

  T x = pow(a, 1.0 / k);
  bool up = true;
  while (!isok(x))
    up = false, x--;
  if (up)
  {
    while (x < numeric_limits<T>::max() && isok(x + 1))
      x++;
  }
  return x;
}

// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back

#define LMD(x,fx) ([&](const auto &x) { return fx; })
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_infty.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_value.hpp

template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}

template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

constexpr array<pll, 4> DRULgrid = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
constexpr array<pll, 4> DRULplane = {{{0, -1}, {1, 0}, {0, 1}, {-1, 0}}};
// https://github.com/miscalculation53/library/tree/wip/template/template_binsearch.hpp

template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;

#define DEFAULT_COMP ranges::less

namespace internal
{
};

// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp

template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }

inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }

inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }

inline constexpr ull lsb_pos(ull x) { assert(x != 0); return countr_zero(x); }
inline constexpr ull msb_pos(ull x) { assert(x != 0); return bit_width(x) - 1; }
inline constexpr ull lsb_mask(ull x) { assert(x != 0); return x & -x; }
inline constexpr ull msb_mask(ull x) { assert(x != 0); return bit_floor(x); }

inline constexpr bool btest(ull x, uint k) { return (x >> k) & 1; }
inline constexpr bool bsubset(ull x, ull y) { return (x & y) == x; }
inline constexpr bool bsupset(ull x, ull y) { return (x & y) == y; }
inline constexpr ull bsetminus(ull x, ull y) { return x & ~y; }
// https://github.com/miscalculation53/library/tree/wip/template/template_inout.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump.hpp

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) 
#define local(...) 
#define oj(...) __VA_ARGS__
#define local_oj(a,b) (b)

namespace fastio {
template <class T>
struct unsigned_integer
{
  using type = make_unsigned_t<T>;
};
template <>
struct unsigned_integer<i128>
{
  using type = u128;
};
template <>
struct unsigned_integer<u128>
{
  using type = u128;
};
template <class T>
using unsigned_integer_t = typename unsigned_integer<T>::type;

static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];

uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd1(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (c <= ' ');
}

void rd1(string &x) {
  x.clear();
  while (true) {
    if (pil == pir) load();
    while (pil < pir && ibuf[pil] <= ' ') ++pil;
    if (pil < pir) break;
  }
  while (true) {
    uint32_t p = pil;
    while (pil < pir && ibuf[pil] > ' ') ++pil;
    x.append(ibuf + p, pil - p);
    if (pil < pir) {
      ++pil;
      return;
    }
    load();
  }
}

template <typename T>
void rd1_real(T &x) {
  string s;
  rd1(s);

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = from_chars(s.data(), s.data() + s.size(), x);
    if (ec == errc{} && p == s.data() + s.size()) return;
  }

  if constexpr (is_same_v<T, long double>)
    x = stold(s);
  else
    x = stod(s);
}

template <bool check_buffer = true, typename T>
void rd1_integer(T &x) {
  using U = unsigned_integer_t<T>;
  bool minus = false;
  U val = 0;
  if constexpr (check_buffer)
    if (pil + 100 > pir) load();
  uint32_t p = pil;
  while (ibuf[p] < '-') ++p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (ibuf[p] == '-') minus = true, ++p;
  }
  while ('0' <= ibuf[p]) val = val * 10 + (ibuf[p++] & 15);
  pil = p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus)
    {
      const U min_abs = U(numeric_limits<T>::max()) + 1;
      x = val == min_abs ? numeric_limits<T>::lowest() : -T(val);
    }
    else x = T(val);
  }
  else
    x = T(val);
}

void rd1(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }

template <class... T>
void read(T &...x) {
  if constexpr (sizeof...(T) <= SIZ / 100 &&
                ((!is_same_v<T, char> &&
                  (is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>)) && ...)) {
    if (pil + 100 * sizeof...(T) > pir) load();
    (rd1_integer<false>(x), ...);
  }
  else
    (rd1(x), ...);
}

void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(string_view s) {
  while (!s.empty()) {
    if (por == SIZ) flush();
    size_t n = min<size_t>(s.size(), SIZ - por);
    memcpy(obuf + por, s.data(), n);
    por += n;
    s.remove_prefix(n);
  }
}

template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  using U = unsigned_integer_t<T>;
  U ux;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (x < 0)
      obuf[por++] = '-', ux = U(0) - U(x);
    else
      ux = U(x);
  }
  else
    ux = x;
  int outi;
  for (outi = 96; ux >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[ux % 10000], 4);
    ux /= 10000;
  }
  if (ux >= 1000) {
    memcpy(obuf + por, pre.num[ux], 4);
    por += 4;
  } else if (ux >= 100) {
    memcpy(obuf + por, pre.num[ux] + 1, 3);
    por += 3;
  } else if (ux >= 10) {
    int q = (ux * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (ux - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = ux | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt1_real(T x) {

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = to_chars(out, out + sizeof(out), x, chars_format::fixed, 15);
    if (ec == errc{}) {
      wt1(string_view(out, p));
      return;
    }
  }

  ostringstream oss;
  oss << fixed << setprecision(15) << x;
  wt1(oss.str());
}

void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }

template <class T, class U>
void wt1(const pair<T, U> &val);
template <class... T>
void wt1(const tuple<T...> &tpl);
template <class T, size_t S>
void wt1(const array<T, S> &val);
template <class T>
void wt1(const vector<T> &val);

template <class... T>
void write(T &&...x) {
  (wt1(std::forward<T>(x)), ...);
}

template <class... T>
void print(T &&...x) {
  if constexpr (sizeof...(T))
  {
    int i = 0;
    ((i++ ? wt1(' ') : void(), wt1(std::forward<T>(x))), ...);
  }
  wt1('\n');
}

} 

struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;

namespace internal
{

template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }

}; 

#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define IN(T,...) T __VA_ARGS__; READ(__VA_ARGS__)

#define CHAR(...) IN(char, __VA_ARGS__)
#define INT(...) IN(int, __VA_ARGS__)
#define LL(...) IN(ll, __VA_ARGS__)
#define STR(...) IN(string, __VA_ARGS__)
#define ARR(T,n,...) array<T, n> __VA_ARGS__; READ(__VA_ARGS__)

#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define READVEC2(...) internal::READVEC2nodump(__VA_ARGS__); dump(__VA_ARGS__)

#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define VEC2(T,n,m,...) vvc<T> __VA_ARGS__; READVEC2(n, m, __VA_ARGS__)

#define READJAG(...) internal::READJAGnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define JAG(T,n,...) vvc<T> __VA_ARGS__; READJAG(n, __VA_ARGS__)

#define ENDL '\n'

#define WRITE fastio::write
#define PRINT fastio::print

#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)

namespace internal
{

}; 

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = zip(tuple{__VA_ARGS__})

// https://github.com/miscalculation53/library/tree/wip/template/template_random.hpp

mt19937_64 mt;

bool randbool(double p)
{
  assert(0 <= p && p <= 1);
  return bernoulli_distribution(p)(mt);
}

namespace internal
{
}; 

// https://github.com/miscalculation53/library/tree/wip/math/modint/template_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_static.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/larger_int.hpp

namespace larger_int_detail
{
}

template <class T>
struct larger_int
{
private:
  static constexpr bool check();
  static_assert(check());

public:
  using type = T;
};

#define LARGER_INT(T,U) template <> struct larger_int<T> { using type = U; };

LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)

LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)

#undef LARGER_INT

template <class T>
struct Rational;

template <class T>
struct larger_int<Rational<T>>
{
  using type = Rational<typename larger_int<T>::type>;
};

template <class T>
using larger_int_t = typename larger_int<T>::type;
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_isprime.hpp

namespace internal
{

template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}

template <auto n>
constexpr bool isprime = isprime_constexpr(n);

};

namespace internal
{

template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = larger_int_t<value_type>;

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_barrett32.hpp

namespace internal
{

struct barrett32
{
  uint m;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_montgomery64.hpp

namespace internal
{

inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}

struct montgomery64odd
{
  ull m, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};

template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{(1LL << 61) - 1};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp

namespace internal
{

  template <class Policy>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;

  private:
    V _v;

  public:

    modint_impl();

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v);

    M val() const;

    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }
    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }
    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }
    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }
    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }
    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }
    friend M safe_hash_key(const mint &x) { return x.val(); }
  
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;

template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;

// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp

template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
  
public:
  inline static vc<T> fac_, finv_, inv_;

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp

// https://github.com/miscalculation53/library/tree/wip/math/svp2d.hpp

namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;

        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };

  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}

using mint = modint998244353;

void init()
{
  oj(mt.seed(random_device()()));
}

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/multiplicative_prefix_sum.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/dirichlet_prefix_sum.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/dirichlet_convolution.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/enumerate_multiplicative.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_basic_ops.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_base.hpp

template <class S_, auto op_, auto e_, auto inv_>
struct Group
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
  static constexpr auto inv = inv_;
};

template <class R>
using GroupOfRingAdd = Group<typename R::S, R::add, R::e0, R::minus>;

template <class Gadd, class Mmul>
struct RingFromGroupMonoid
{
  static_assert(is_same_v<typename Gadd::S, typename Mmul::S>, "Gadd::S and Mmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

template <class Gadd, class Gmul>
struct FieldFromGroupGroup
{
  static_assert(is_same_v<typename Gadd::S, typename Gmul::S>, "Gadd::S and Gmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Gmul::op;
  static constexpr auto e1 = Gmul::e;
  static constexpr auto inv = Gmul::inv;
};

template <class T, class = void>
struct has_e1 : false_type {};
template <class T>
struct has_e1<T, void_t<decltype(T::e1())>> : true_type {};
template <class T>
inline constexpr bool has_e1_v = has_e1<T>::value;

template <class T>
struct MonoidMul
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e()
  {
    if constexpr (has_e1_v<S>)
      return S::e1();
    else
      return 1;
  }
};

template <class T>
struct GroupAddSub
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return S{};
  }
  static constexpr S inv(S a) { return -a; }
  template <class I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k) { return a * k; }
};

template <class T>
using RingAddSubMul = RingFromGroupMonoid<GroupAddSub<T>, MonoidMul<T>>;
// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/linear_sieve.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/prime_power.hpp

template <class P>
struct PrimePower
{
  P p;
  int e;
  P pe;

  PrimePower() : p(-1), e(-1), pe(-1) {}
  PrimePower(P p, int e, P pe) : p(p), e(e), pe(pe) {}

};

tuple<int, ll, ll> ord_pow_div(ll n, ll m)
{
  assert(m >= 2);
  if (m == 2)
  {
    int e = countr_zero(n);
    return {e, 1LL << e, n >> e};
  }
  if (n % m != 0)
    return {0, 1, n};
  n /= m;
  if (n % m != 0)
    return {1, m, n};
  n /= m;
  ll m2 = m * m;
  auto [f, m2f, nn] = ord_pow_div(n, m2);
  int e = 2 + 2 * f;
  ll me = m2f * m2;
  if (nn % m == 0)
    e++, me *= m, nn /= m;
  return {e, me, nn};
}

template <class P>
vc<PrimePower<P>> factorized_mul
(const vc<PrimePower<P>> &fac1, const vc<PrimePower<P>> &fac2)
{
  const int n = fac1.size(), m = fac2.size();
  vc<PrimePower<P>> fac;
  fac.reserve(n + m);
  int i = 0, j = 0;
  while (i < n && j < m)
  {
    if (fac1[i].p < fac2[j].p)
      fac.emplace_back(fac1[i++]);
    else if (fac1[i].p > fac2[j].p)
      fac.emplace_back(fac2[j++]);
    else
    {
      using U = larger_int_t<P>;
      fac.emplace_back(fac1[i].p, fac1[i].e + fac2[j].e,
                       U(fac1[i].pe) * U(fac2[j].pe));
      i++, j++;
    }
  }
  fac.insert(fac.end(), fac1.begin() + i, fac1.end());
  fac.insert(fac.end(), fac2.begin() + j, fac2.end());
  return fac;
}

struct LinearSieve
{
public:
  static int n;
  static vc<PrimePower<int>> lpf_;
  static vc<int> primes;

  static void reserve(int n_)
  {
    if (n_ <= n)
      return;
    n = max(n_, 2 * n);
    lpf_.resize(n + 1);
    for (int d = 2; d <= n; d++)
    {
      if (lpf_[d].p == -1)
      {
        lpf_[d] = PrimePower<int>(d, 1, d);
        primes.eb(d);
      }
      fec(p : primes)
      {
        if (p > n / d || p > lpf_[d].p)
          break;
        if (lpf_[d].p == p)
          lpf_[p * d] = PrimePower<int>(p, lpf_[d].e + 1, lpf_[d].pe * p);
        else
          lpf_[p * d] = PrimePower<int>(p, 1, p);
      }
    }
  }

  template <class P = int>
  static PrimePower<P> lpf(int n)
  {
    assert(n >= 1);
    reserve(n);
    return lpf_[n];
  }

  static bool is_prime(int n)
  {
    if (n <= 1)
      return false;
    return lpf(n).p == n;
  }

  static int Omega(int n)
  {
    assert(1 <= n);
    static vc<unsigned char> table{0, 0};
    if (n >= int(table.size()))
    {
      reserve(n);
      const int first = int(table.size());
      table.resize(n + 1);
      for (int i = first; i <= n; i++)
        table[i] = table[i / lpf_[i].p] + 1;
    }
    return table[n];
  }

};
vc<PrimePower<int>> LinearSieve::lpf_{};
int LinearSieve::n{};
vc<int> LinearSieve::primes{};

namespace internal
{
  
  template <class M>
  typename M::S multiplicative_monoid_power(typename M::S a, ll k)
  {
    assert(k >= 0);
    if constexpr (HasMonoidPow<M, ll>::value)
      return M::pow(a, k);
    else
    {
      auto res = M::e();
      while (k > 0)
      {
        if (k & 1)
          res = M::op(res, a);
        k >>= 1;
        if (k > 0)
          a = M::op(a, a);
      }
      return res;
    }
  }

  template <class G>
  typename G::S multiplicative_integer_multiple(typename G::S a, ll n)
  {
    if (n < 0)
    {
      a = G::inv(a);
      return G::op(multiplicative_monoid_power<G>(a, -(n + 1)), a);
    }
    return multiplicative_monoid_power<G>(a, n);
  }

  template <class R>
  struct multiplicative_integer_embedding
  {
  };
  
  template <class G, class M>
  struct multiplicative_integer_embedding<RingFromGroupMonoid<G, M>>
  {
    static typename G::S get(ll n)
    { return multiplicative_integer_multiple<G>(M::e(), n); }
  };
  template <class G, class H>
  struct multiplicative_integer_embedding<FieldFromGroupGroup<G, H>>
  {
  };
  template <class R>
  typename R::S multiplicative_from_integer(ll n)
  { return multiplicative_integer_embedding<R>::get(n); }

  template <class R, class F, class Q>
  decltype(auto) eval_primepower(const F &f, const Q &q)
  {
    if constexpr (is_invocable_v<const F &, const Q &>)
      return f(q);
    else
      return f(q, R{});
  }
}

inline constexpr auto mobius_primepower = [](const auto &q, auto ring)
{
  using R = decltype(ring);
  return q.e == 0 ? R::e1() : q.e == 1 ? R::minus(R::e1()) : R::e0();
};

// https://github.com/miscalculation53/library/tree/wip/math/modint/inv_many.hpp

namespace internal
{

}

template <class R>
struct DirichletSeries
{
  using S = typename R::S;

private:
  int n_;
  vc<S> f_;
  bool multiplicative_;

public:

  friend DirichletSeries operator*(const S &a, const DirichletSeries &f) { return f * a; }

private:
  
public:
  
};

inline constexpr auto e_prefix_sum = [](ll n, auto ring)
{ return n == 0 ? decltype(ring)::e0() : decltype(ring)::e1(); };
inline constexpr auto zeta_prefix_sum = [](ll n, auto ring)
{ return internal::multiplicative_from_integer<decltype(ring)>(n); };

namespace internal
{
  
  template <int D>
  struct dirichlet_root_prefix_sum_index
  {
    static_assert(D >= 1);
    ll n, maximum;
    int k, large;
    vc<ll> large_value;
    vc<int> raw_index;

  };

  template <>
  struct dirichlet_root_prefix_sum_index<1>
  {
    ll n;
    int k, large;
    explicit dirichlet_root_prefix_sum_index(ll n) : n(n)
    {
      assert(n >= 0);
      const ll root = iroot(n, 2), l = n / (root + 1);
      k = int(root), large = int(l);
    }
    int size() const { return k + large; }
    int index(ll v) const { return v <= k ? int(v) - 1 : size() - int(n / v); }
    ll value(int i) const { return i < k ? ll(i) + 1 : n / (size() - i); }
  };
  using dirichlet_prefix_sum_index = dirichlet_root_prefix_sum_index<1>;

  template <class R, class GetF>
  typename R::S eval_dirichlet_prefix(const GetF &getF, ll n)
  {
    if constexpr (is_invocable_v<const GetF &, ll>)
      return getF(n);
    else
      return getF(n, R{});
  }
}

template <class R, int D = 1>
struct DirichletPrefixSum
{
  using S = typename R::S;

private:
  ll n_;
  int k_, l_;
  vc<S> small_, large_;
  bool multiplicative_;
  internal::dirichlet_root_prefix_sum_index<D> coordinates_;

  struct same_shape {};
  DirichletPrefixSum(const DirichletPrefixSum &a, same_shape)
    : n_(a.n_), k_(a.k_), l_(a.l_), small_(k_ + 1, R::e0()),
      large_(l_ + 1, R::e0()), multiplicative_(false), coordinates_(a.coordinates_) {}

  int large_index(ll x) const
  {
    if constexpr (D == 1) return int(n_ / x);
    else return coordinates_.raw_index[size_t(n_ / ipow<ll>(x, D))];
  }

public:
  DirichletPrefixSum() : DirichletPrefixSum(0) {}

  explicit DirichletPrefixSum(ll n)
    : n_(n), multiplicative_(false), coordinates_(n)
  {
    k_ = coordinates_.k, l_ = coordinates_.large;
    small_.assign(k_ + 1, R::e0());
    large_.assign(l_ + 1, R::e0());
  }

  template <class GetF, enable_if_t<is_invocable_v<const GetF &, ll>
    || is_invocable_v<const GetF &, ll, R>, int> = 0>
  DirichletPrefixSum(ll n, const GetF &getF, bool multiplicative = false)
    : DirichletPrefixSum(n)
  {
    for (int x = 1; x <= k_; x++)
      small_[x] = internal::eval_dirichlet_prefix<R>(getF, x);
    for (int i = 1; i <= l_; i++)
      large_[i] = internal::eval_dirichlet_prefix<R>(getF, value(size() - i));
    multiplicative_ = multiplicative;
    assert(!multiplicative || n == 0 || small_[1] == R::e1());
  }

  static DirichletPrefixSum unit(ll n)
  { return DirichletPrefixSum(n, e_prefix_sum, true); }

  ll n() const { return n_; }
  int size() const { return k_ + l_; }
  ll value(int i) const
  {
    assert(0 <= i && i < size());
    if constexpr (D == 1) return i < k_ ? ll(i) + 1 : n_ / (size() - i);
    else return i < k_ ? ll(i) + 1 : coordinates_.large_value[size() - i];
  }
  
  bool contains(ll x) const
  {
    if constexpr (D == 1)
      return 0 <= x && x <= n_ && (x <= k_ || n_ / (n_ / x) == x);
    else
      return 0 <= x && x <= coordinates_.maximum
        && (x <= k_ || coordinates_.large_value[large_index(x)] == x);
  }
  const S &F(ll x) const
  {
    assert(contains(x));
    return x <= k_ ? small_[size_t(x)] : large_[large_index(x)];
  }
  
  void setF(ll x, const S &value)
  {
    assert(x >= 1 && contains(x));
    (x <= k_ ? small_[size_t(x)] : large_[large_index(x)]) = value;
    multiplicative_ = false;
  }

private:

public:
  DirichletPrefixSum operator-() const
  {
    DirichletPrefixSum res(*this);
    for (int i = 1; i <= k_; i++)
      res.small_[i] = R::minus(res.small_[i]);
    for (int i = 1; i <= l_; i++)
      res.large_[i] = R::minus(res.large_[i]);
    res.multiplicative_ = false;
    return res;
  }
  friend DirichletPrefixSum operator*(const S &a, const DirichletPrefixSum &f) { return f * a; }

private:
  
public:
  
};

// https://github.com/miscalculation53/library/tree/wip/ds/fenwick_tree/fenwick_tree.hpp

template <class G>
struct FenwickTree
{
  using S = typename G::S;

private:
  int n;
  vc<S> dat;

public:
  FenwickTree() {}
  FenwickTree(int n) : n(n), dat(n + 1, G::e()) {}

  S sum(int r) const
  {
    assert(0 <= r && r <= n);
    S s = G::e();
    while (r > 0)
    {
      s = G::op(s, dat[r]);
      r -= r & -r;
    }
    return s;
  }
  
  void add(int i, S x)
  {
    assert(0 <= i && i < n);
    i++;
    while (i <= n)
    {
      dat[i] = G::op(dat[i], x);
      i += i & -i;
    }
  }
  
};

namespace internal
{
  template <class R>
  DirichletPrefixSum<R> prime_prefix_sum_sieve(const DirichletPrefixSum<R> &a)
  {
    const ll n = a.n();
    const int root = int(iroot(n, 2));
    assert(n == 0 || a.F(1) == R::e1());
    LinearSieve::reserve(root);
    DirichletPrefixSum<R> dp(n, [&](ll x) { return R::add(a.F(x), R::minus(R::e1())); });
    for (int p : LinearSieve::primes)
    {
      if (p > root) break;
      const auto fp = R::add(a.F(p), R::minus(a.F(p - 1))), before = dp.F(p - 1);
      auto update = [&](ll x)
      {
        dp.setF(x, R::add(dp.F(x), R::minus(R::mul(fp,
          R::add(dp.F(x / p), R::minus(before))))));
      };
      for (ll j = 1; j <= n / (ll(root) + 1) && n / j >= ll(p) * p; j++) update(n / j);
      for (ll x = root; x >= ll(p) * p; x--) update(x);
    }
    return dp;
  }

  template <class R, class GetPrimePower>
  DirichletPrefixSum<R> multiplicative_prefix_sum_sieve(
    const DirichletPrefixSum<R> &prime_sum, const GetPrimePower &f_primepower)
  {
    using S = typename R::S;
    const ll n = prime_sum.n();
    const int root = int(iroot(n, 2));
    assert(n == 0 || prime_sum.F(1) == R::e0());
    LinearSieve::reserve(root);
    DirichletPrefixSum<R> res(n, [&](ll x) { return prime_sum.F(x); });
    for (int pi = int(upper_bound(LinearSieve::primes.begin(), LinearSieve::primes.end(), root)
                      - LinearSieve::primes.begin()) - 1; pi >= 0; pi--)
    {
      const int p = LinearSieve::primes[pi];
      vc<pair<ll, S>> powers;
      for (ll q = p, e = 1;; q *= p, e++)
      {
        powers.emplace_back(q, internal::eval_primepower<R>(f_primepower, PrimePower<ll>(p, int(e), q)));
        if (q > n / p) break;
      }
      const S before = prime_sum.F(p);
      auto update = [&](ll x)
      {
        S value = res.F(x);
        for (size_t e = 0; e + 1 < powers.size() && powers[e + 1].first <= x; e++)
        {
          value = R::add(value, R::mul(powers[e].second,
            R::add(res.F(x / powers[e].first), R::minus(before))));
          value = R::add(value, powers[e + 1].second);
        }
        res.setF(x, value);
      };
      for (ll j = 1; j <= n / (ll(root) + 1) && n / j >= ll(p) * p; j++) update(n / j);
      for (ll x = root; x >= ll(p) * p; x--) update(x);
    }
    return DirichletPrefixSum<R>(n, [&](ll x) { return R::add(res.F(x), R::e1()); }, true);
  }

  template <class R>
  struct multiplicative_sieve_updates
  {
    ll n, bound;
    int root, large, top;
    dirichlet_prefix_sum_index coordinates;
    FenwickTree<GroupOfRingAdd<R>> bit;

    explicit multiplicative_sieve_updates(ll n)
      : n(n), root(int(iroot(n, 2))), large(int(n / (ll(root) + 1))),
        top(min(large, int(iroot(n, 3)))), coordinates(n)
    {
      bound = n / (ll(top) + 1);
      bit = FenwickTree<GroupOfRingAdd<R>>(root + large - top);
    }
    int index(ll x) const
    { return coordinates.index(x); }
    void add(ll v, const typename R::S &value)
    {
      assert(1 <= v && v <= bound);
      if (value != R::e0()) bit.add(index(v), value);
    }
    typename R::S operator[](ll x) const
    {
      assert(0 <= x && x <= bound);
      return bit.sum(index(x) + 1);
    }
    template <class Apply>
    void apply(DirichletPrefixSum<R> &dp, const Apply &op) const
    {
      for (int x = 1; x <= root; x++) dp.setF(x, op(dp.F(x), (*this)[x]));
      for (int j = large; j > top; j--) dp.setF(n / j, op(dp.F(n / j), (*this)[n / j]));
    }
  };

  template <class R>
  DirichletPrefixSum<R> prime_prefix_sum_sieve_2_3(const DirichletPrefixSum<R> &a)
  {
    using S = typename R::S;
    const ll n = a.n();
    if (n == 0) return {};
    assert(a.F(1) == R::e1());
    const int root = int(iroot(n, 2)), cut = int(iroot(n, 6)), cube = int(iroot(n, 3));
    LinearSieve::reserve(root);
    vc<S> weight(root + 1, R::e0());
    for (int p : LinearSieve::primes) { if (p > root) break; weight[p] = R::add(a.F(p), R::minus(a.F(p - 1))); }
    DirichletPrefixSum<R> dp(n, [&](ll x) { return R::add(a.F(x), R::minus(R::e1())); });
    internal::multiplicative_sieve_updates<R> delta(n);
    const auto sub = [](const S &x, const S &y) { return R::add(x, R::minus(y)); };
    auto lucy = [&](int p)
    {
      const S before = dp.F(p - 1);
      auto update = [&](ll x) { dp.setF(x, sub(dp.F(x), R::mul(weight[p], sub(dp.F(x / p), before)))); };
      for (int j = 1; j <= delta.large && n / j >= ll(p) * p; j++) update(n / j);
      for (ll x = root; x >= ll(p) * p; x--) update(x);
    };
    size_t pi = 0;
    while (pi < LinearSieve::primes.size() && LinearSieve::primes[pi] <= cut) lucy(LinearSieve::primes[pi++]);
    for (; pi < LinearSieve::primes.size() && LinearSieve::primes[pi] <= cube; pi++)
    {
      const int p = LinearSieve::primes[pi];
      const S before = dp.F(p - 1);
      for (int j = 1; j <= delta.top && n / j >= ll(p) * p; j++)
      {
        const ll x = n / j, y = x / p;
        S value = dp.F(y);
        if (y <= delta.bound) value = sub(value, delta[y]);
        dp.setF(x, sub(dp.F(x), R::mul(weight[p], sub(value, before))));
      }
      auto dfs = [&](auto &&self, ll v, size_t first, const S &f) -> void
      {
        if (v != p) delta.add(v, f);
        for (size_t j = first; j < LinearSieve::primes.size(); j++)
        {
          const int q = LinearSieve::primes[j];
          if (q > delta.bound / v) break;
          self(self, v * q, j, R::mul(f, weight[q]));
        }
      };
      dfs(dfs, p, pi, weight[p]);
    }
    delta.apply(dp, sub);
    while (pi < LinearSieve::primes.size() && LinearSieve::primes[pi] <= root) lucy(LinearSieve::primes[pi++]);
    return dp;
  }

  template <class R, class GetPrimePower>
  DirichletPrefixSum<R> multiplicative_prefix_sum_sieve_2_3(
    const DirichletPrefixSum<R> &prime_sum, const GetPrimePower &f_primepower)
  {
    using S = typename R::S;
    const ll n = prime_sum.n();
    if (n == 0) return DirichletPrefixSum<R>::unit(0);
    assert(prime_sum.F(1) == R::e0());
    const int root = int(iroot(n, 2)), cut = int(iroot(n, 6)), cube = int(iroot(n, 3));
    LinearSieve::reserve(root);
    const auto &primes = LinearSieve::primes;
    const size_t begin = upper_bound(primes.begin(), primes.end(), cube) - primes.begin();
    vc<S> fp(root + 1, R::e0()), fp2(fp);
    for (int p : primes)
    {
      if (p > root) break;
      fp[p] = internal::eval_primepower<R>(f_primepower, PrimePower<ll>(p, 1, p));
      fp2[p] = internal::eval_primepower<R>(f_primepower, PrimePower<ll>(p, 2, ll(p) * p));
    }
    const auto sub = [](const S &x, const S &y) { return R::add(x, R::minus(y)); };
    DirichletPrefixSum<R> dp(n, [&](ll x)
    {
      S value = R::e1();
      if (x <= cube) return value;
      value = R::add(value, sub(prime_sum.F(x), prime_sum.F(cube)));
      for (size_t j = begin; j < primes.size() && primes[j] <= x / primes[j]; j++)
      {
        const int p = primes[j];
        value = R::add(value, R::add(fp2[p], R::mul(fp[p], sub(prime_sum.F(x / p), prime_sum.F(p)))));
      }
      return value;
    });
    internal::multiplicative_sieve_updates<R> delta(n);
    for (size_t pi = begin; pi > 0;)
    {
      const int p = primes[--pi];
      vc<pair<ll, S>> powers;
      for (ll q = p, e = 1;; q *= p, e++)
      {
        powers.emplace_back(q, internal::eval_primepower<R>(f_primepower, PrimePower<ll>(p, int(e), q)));
        if (q > n / p) break;
      }
      auto update = [&](ll x)
      {
        S value = dp.F(x);
        for (const auto &[q, f] : powers)
        {
          if (q > x) break;
          const ll y = x / q;
          S previous = dp.F(y);
          if (p > cut && y <= delta.bound) previous = R::add(previous, delta[y]);
          value = R::add(value, R::mul(f, previous));
        }
        dp.setF(x, value);
      };
      if (p > cut)
      {
        for (int j = 1; j <= delta.top; j++) update(n / j);
        auto dfs = [&](auto &&self, ll v, size_t i, int e, ll pe, const S &other) -> void
        {
          const int q = primes[i];
          const S f = R::mul(other, e == 1 ? fp[q] :
            internal::eval_primepower<R>(f_primepower, PrimePower<ll>(q, e, pe)));
          delta.add(v, f);
          if (q <= delta.bound / v) self(self, v * q, i, e + 1, pe * q, other);
          for (size_t j = i + 1; j < primes.size() && primes[j] <= delta.bound / v; j++)
            self(self, v * primes[j], j, 1, primes[j], f);
        };
        dfs(dfs, p, pi, 1, p, R::e1());
      }
      else
      {
        if (pi + 1 == begin || primes[pi + 1] > cut) delta.apply(dp, R::add);
        for (int j = 1; j <= delta.large; j++) update(n / j);
        for (int x = root; x >= p; x--) update(x);
      }
    }
    if (cut < 2) delta.apply(dp, R::add);
    return DirichletPrefixSum<R>(n, [&](ll x) { return dp.F(x); }, true);
  }

  inline constexpr ll multiplicative_prefix_sum_fenwick_threshold = 1'000'000;
} 

template <class R>
DirichletPrefixSum<R> prime_prefix_sum(const DirichletPrefixSum<R> &g)
{
  return g.n() < internal::multiplicative_prefix_sum_fenwick_threshold
    ? internal::prime_prefix_sum_sieve(g) : internal::prime_prefix_sum_sieve_2_3(g);
}

template <class R, class GetPrimePower>
DirichletPrefixSum<R> multiplicative_prefix_sum(const DirichletPrefixSum<R> &f_prime,
  const GetPrimePower &f_primepower)
{
  return f_prime.n() < internal::multiplicative_prefix_sum_fenwick_threshold
    ? internal::multiplicative_prefix_sum_sieve(f_prime, f_primepower)
    : internal::multiplicative_prefix_sum_sieve_2_3(f_prime, f_primepower);
}

void main2()
{
  LL(N);

  DirichletPrefixSum<RingAddSubMul<ll>> zeta(N, zeta_prefix_sum);
  auto pi = prime_prefix_sum(zeta);
  auto mu = multiplicative_prefix_sum(-pi, mobius_primepower);
  PRINT(mu.F(N));
}

void test()
{

}

// https://github.com/miscalculation53/library/tree/wip/template/template_main.hpp

template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      
    };
  
    CERR("\n[FAST_IO]\n\n", "32");
    
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}