結果
| 問題 | No.3691 Calculate Mu Sum |
| コンテスト | |
| ユーザー |
 miscalc
|
| 提出日時 | 2026-09-11 05:20:40 |
| 言語 | C++23 (gcc 15.3.0 + boost 1.92.0 + ACL) |
| 結果 |
AC
不安定
|
| 実行時間 | 2 ms / 2,000 ms |
| + 416µs | |
| コード長 | 55,494 bytes |
| 記録 | |
| コンパイル時間 | 3,358 ms |
| コンパイル使用メモリ | 370,052 KB |
| 実行使用メモリ | 6,400 KB |
| 最終ジャッジ日時 | 2026-09-11 05:21:05 |
| 合計ジャッジ時間 | 5,001 ms |
|
ジャッジサーバーID （参考情報） |
judge1_0 / judge3_0 |
（要ログイン）
| 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, class = void>
struct has_infty : false_type {};
template <class T>
struct has_infty<T, void_t<decltype(T::infty())>> : true_type {};
template <class T>
const T &custom_value()
{
static const T value = T::infty();
return value;
}
template <class T>
inline constexpr bool unsupported = false;
}
template <class T>
constexpr decltype(auto) default_infty()
{
if constexpr (default_infty_detail::has_infty<T>::value)
return default_infty_detail::custom_value<T>();
else if constexpr (is_same_v<T, i128> || is_same_v<T, u128>)
return T(INF) * T(INF);
else if constexpr (is_integral_ext<T> && !is_same_v<T, bool>)
{
if constexpr (sizeof(T) >= sizeof(ll))
return T(INF);
else if constexpr (numeric_limits<T>::digits >= 31)
return (T(1) << 30) - 1;
else
return T(numeric_limits<T>::max() / 2);
}
else if constexpr (numeric_limits<T>::has_infinity)
return numeric_limits<T>::infinity();
else
static_assert(default_infty_detail::unsupported<T>, "No default infinity for this type; specify infty explicitly or define T::infty().");
}
template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : 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 divround(U a, V b) { return divfloor<T>(2 * T(a) + T(b), 2 * T(b)); }
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)
{
assert(b >= 0);
if (b == 0)
return 1;
if (a == 0 || a == 1)
return a;
if (a < 0 && a == -1)
return b & 1 ? -1 : 1;
T res = 1, tmp = a;
while (true)
{
if (b & 1)
res *= tmp;
b >>= 1;
if (b == 0)
break;
tmp *= tmp;
}
return res;
}
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, auto x, enable_if_t<!is_invocable_v<decltype(x)>, int> = 0>
constexpr T resolved_value()
{
return T(x);
}
template <class T, auto x, enable_if_t<is_invocable_v<decltype(x)>, long> = 0>
const T &resolved_value()
{
static const T value = T(x());
return value;
}
template <class T, auto x = nullptr>
constexpr decltype(auto) resolved_infty()
{
if constexpr (is_same_v<decltype(x), nullptr_t>)
return default_infty<T>();
else
return resolved_value<T, x>();
}
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>
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 mod_type mod() { return M; }
static constexpr value_type umod() { return M; }
static constexpr value_type init(value_type v) { return v; }
static constexpr mod_type val(value_type v) { return v; }
static constexpr value_type mul(value_type a, value_type b)
{
return (value_type)((calc_type(a) * b) % M);
}
};
};
// 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 mod_type mod() { return reducer.umod(); }
static value_type umod() { return reducer.umod(); }
static value_type init(value_type v) { return v; }
static mod_type val(value_type v) { return v; }
static value_type mul(value_type a, value_type b) { return reducer.mul(a, b); }
};
};
// 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 mod_type mod() { return reducer.umod(); }
static value_type umod() { return reducer.umod(); }
static value_type init(value_type v) { return reducer.inv_reduce(v); }
static mod_type val(value_type v) { return reducer.reduce(v); }
static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};
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 mod_type mod() { return reducer.umod(); }
static value_type umod() { return reducer.umod(); }
static value_type init(value_type v) { return reducer.inv_reduce(v); }
static mod_type val(value_type v) { return reducer.reduce(v); }
static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};
};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp
template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
if (a == 0 && b == 0)
return {0, 0, 0};
T x1 = 1, y1 = 0, z1 = a;
T x2 = 0, y2 = 1, z2 = b;
while (z2 != 0)
{
T q = z1 / z2;
tie(x1, x2) = make_pair(x2, x1 - q * x2);
tie(y1, y2) = make_pair(y2, y1 - q * y2);
tie(z1, z2) = make_pair(z2, z1 - q * z2);
}
if (z1 < 0)
x1 = -x1, y1 = -y1, z1 = -z1;
return {z1, x1, y1};
}
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:
static constexpr M mod() { return Policy::mod(); }
static mint raw(V v)
{
mint x;
x._v = v;
return x;
}
M val() const { return Policy::val(_v); }
template <class T>
mint pow(T n) const
{
assert(n >= 0);
mint x = *this, r = 1;
while (n)
{
if (n & 1)
r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const
{
if constexpr (Policy::is_prime)
{
return pow(mod() - 2);
}
else
{
auto [g, x, y] = extgcd<M>(val(), mod());
assert(g == 1);
return mint(x);
}
}
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;
template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};
// 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_;
static void reserve(int n)
{
if constexpr (is_dynamic_modint_v<T>)
{
if (mod != T::mod())
{
mod = T::mod();
fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
}
}
else
{
if (fac_.empty())
fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
}
if (n < SZ(fac_))
return;
chmin(n, T::mod() - 1);
int si = fac_.size();
fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
repi(i, si, n + 1)
{
fac_[i] = fac_[i - 1] * T::raw(i);
inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
finv_[i] = finv_[i - 1] * inv_[i];
}
}
static T inv(int n)
{
n %= T::mod();
if (n < 0)
n += T::mod();
assert(n != 0);
reserve(n);
return inv_[n];
}
};
// 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
template <class T = ll, class U = larger_int_t<T>>
pair<T, T> svp2d(const pair<T, T> &a, const pair<T, T> &b)
{
assert((a != pair<T, T>{0, 0} && b != pair<T, T>{0, 0}));
auto [a1, a2] = a;
auto [b1, b2] = b;
if ((U)a1 * a1 + (U)a2 * a2 < (U)b1 * b1 + (U)b2 * b2)
swap(a1, b1), swap(a2, b2);
while ((U)a1 * a1 + (U)a2 * a2 > (U)b1 * b1 + (U)b2 * b2)
{
swap(a1, b1), swap(a2, b2);
T k = divround<U>((U)a1 * b1 + (U)a2 * b2, (U)a1 * a1 + (U)a2 * a2);
b1 -= k * a1, b2 -= k * a2;
if (b1 == 0 && b2 == 0)
return {a1, a2};
}
return {a1, a2};
}
template <class mint>
pair<decltype(mint(0).val()), decltype(mint(0).val())>
mint_to_rat(const mint &x)
{
auto [p, q] = svp2d({x.val(), 1}, {mint::mod(), 0});
if (q < 0)
p = -p, q = -q;
return {p, q};
}
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>
struct GroupMulDiv
{
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 S(1);
}
static constexpr S inv(S a) { return e() / a; }
};
template <class T>
using RingAddSubMul = RingFromGroupMonoid<GroupAddSub<T>, MonoidMul<T>>;
template <class T>
using FieldAddSubMulDiv = FieldFromGroupGroup<GroupAddSub<T>, GroupMulDiv<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 = 1) : p(p), e(e), pe(ipow(p, e)) {}
PrimePower(P p, int e, P pe) : p(p), e(e), pe(pe) {}
template <class P2>
PrimePower(const PrimePower<P2> &pp) : p(pp.p), e(pp.e), pe(pp.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
{
static typename R::S get(ll n)
{ return multiplicative_integer_multiple<GroupOfRingAdd<R>>(R::e1(), n); }
};
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>>
{
static typename G::S get(ll n)
{ return multiplicative_integer_multiple<G>(H::e(), n); }
};
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{});
}
}
template <class R, class F>
vc<typename R::S> enumerate_multiplicative(int n, const F &f_primepower)
{
assert(0 <= n);
LinearSieve::reserve(n);
vc<typename R::S> res(n + 1, R::e0());
if (n >= 1)
res[1] = R::e1();
for (int d = 2; d <= n; d++)
{
const auto q = LinearSieve::lpf_[d];
if (d == q.pe)
res[d] = internal::eval_primepower<R>(f_primepower, q);
else
res[d] = R::mul(res[d / q.pe], res[q.pe]);
}
return res;
}
template <class F>
auto enumerate_multiplicative(int n, const F &f_primepower)
{
using S = decay_t<invoke_result_t<const F &, const PrimePower<int> &>>;
return enumerate_multiplicative<RingAddSubMul<S>>(n, f_primepower);
}
template <class R, class F>
vc<typename R::S> enumerate_completely_multiplicative(int n, const F &f_primepower)
{
assert(0 <= n);
LinearSieve::reserve(n);
vc<typename R::S> res(n + 1, R::e0());
if (n >= 1)
res[1] = R::e1();
for (int d = 2; d <= n; d++)
{
const int p = LinearSieve::lpf_[d].p;
if (d == p)
res[d] = internal::eval_primepower<R>(f_primepower, PrimePower<int>(p, 1, p));
else
res[d] = R::mul(res[p], res[d / p]);
}
return res;
}
template <class F>
auto enumerate_completely_multiplicative(int n, const F &f_primepower)
{
using S = decay_t<invoke_result_t<const F &, const PrimePower<int> &>>;
return enumerate_completely_multiplicative<RingAddSubMul<S>>(n, f_primepower);
}
// https://github.com/miscalculation53/library/tree/wip/math/modint/inv_many.hpp
template <class R>
vc<typename R::S> inv_many(const vc<typename R::S> &a)
{
const int n = a.size();
if (n == 0) return {};
vc<typename R::S> p(n + 1, R::e1());
repi(i, n) p[i + 1] = R::mul(p[i], a[i]);
auto ip = R::inv(p.back());
assert(R::mul(ip, p.back()) == R::e1());
auto res = a;
repi(i, n - 1, -1, -1)
{
res[i] = R::mul(ip, p[i]);
ip = R::mul(ip, a[i]);
}
return res;
}
namespace internal
{
template <class R, class = void>
struct dirichlet_has_inv : false_type {};
template <class R>
struct dirichlet_has_inv<R, void_t<decltype(R::inv(declval<const typename R::S &>()))>> : true_type {};
template <class R>
struct dirichlet_modint_field : false_type {};
template <class S>
struct dirichlet_modint_field<FieldAddSubMulDiv<S>> : is_modint<S> {};
template <class R>
vc<typename R::S> dirichlet_integers(int n)
{
vc<typename R::S> res(n + 1, R::e0());
for (int i = 1; i <= n; i++)
res[i] = R::add(res[i - 1], R::e1());
return res;
}
template <class R>
const vc<typename R::S> &dirichlet_integer_inverses(int n)
{
static_assert(dirichlet_has_inv<R>::value, "R::inv is required");
using S = typename R::S;
assert(n >= 0);
static vc<S> inverse{R::e0(), R::e1()};
static S last_integer = R::e1();
if constexpr (has_mod<R>::value || has_mod<S>::value)
{
const auto current_mod = []
{
if constexpr (has_mod<R>::value) return R::mod();
else return S::mod();
}();
static auto cached_mod = current_mod;
if (cached_mod != current_mod)
{
cached_mod = current_mod;
inverse = {R::e0(), R::e1()};
last_integer = R::e1();
}
}
if (n < int(inverse.size())) return inverse;
vc<S> added;
added.reserve(n + 1 - inverse.size());
S current = last_integer;
for (int i = int(inverse.size()); i <= n; i++)
{
current = R::add(current, R::e1());
assert(current != R::e0());
added.push_back(current);
}
const auto added_inverse = inv_many<R>(added);
inverse.insert(inverse.end(), added_inverse.begin(), added_inverse.end());
last_integer = current;
return inverse;
}
template <class R>
struct dirichlet_divisor
{
using S = typename R::S;
static constexpr bool exact_integer = is_integral_ext<S> && is_same_v<R, RingAddSubMul<S>>;
S value;
bool identity;
explicit dirichlet_divisor(const S &a) : value(a), identity(a == R::e1())
{
if (identity)
return;
assert(a != R::e0());
if constexpr (dirichlet_has_inv<R>::value)
value = R::inv(a);
else if constexpr (!exact_integer)
assert(a == R::minus(R::e1()));
}
S operator()(const S &a) const
{
if (identity)
return a;
if constexpr (dirichlet_has_inv<R>::value)
return R::mul(a, value);
else if constexpr (exact_integer)
return a / value;
else
return R::minus(a);
}
};
}
template <class R>
struct DirichletSeries
{
using S = typename R::S;
private:
int n_;
vc<S> f_;
bool multiplicative_;
public:
explicit DirichletSeries(int n = 0) : n_(n), multiplicative_(false)
{
assert(0 <= n);
f_.assign(n + 1, R::e0());
}
explicit DirichletSeries(vc<S> values, bool multiplicative = false)
: n_(0), f_(move(values)), multiplicative_(multiplicative)
{
if (f_.empty())
f_.push_back(R::e0());
n_ = int(f_.size()) - 1;
f_[0] = R::e0();
assert(!multiplicative_ || n_ == 0 || f_[1] == R::e1());
}
template <class F>
DirichletSeries(int n, const F &f_primepower, bool completely_multiplicative = false)
: DirichletSeries(completely_multiplicative
? enumerate_completely_multiplicative<R>(n, f_primepower)
: enumerate_multiplicative<R>(n, f_primepower), true) {}
static DirichletSeries unit(int n)
{
DirichletSeries res(n);
if (n >= 1)
res.f_[1] = R::e1();
res.multiplicative_ = true;
return res;
}
const S &f(int i) const
{
assert(0 <= i && i <= n_);
return f_[i];
}
friend DirichletSeries operator*(const S &a, const DirichletSeries &f) { return f * a; }
private:
public:
DirichletSeries inv() const { return unit(n_) / *this; }
DirichletSeries diff() const
{
DirichletSeries res(n_);
if (n_ <= 1) return res;
LinearSieve::Omega(n_);
const auto integers = internal::dirichlet_integers<R>(int(msb_pos(n_)));
for (int i = 2; i <= n_; i++)
res.f_[i] = R::mul(integers[LinearSieve::Omega(i)], f_[i]);
return res;
}
DirichletSeries integ(const S &constant = R::e0()) const
{
static_assert(internal::dirichlet_has_inv<R>::value, "R::inv is required");
DirichletSeries res(n_);
if (n_ == 0) return res;
assert(f_[1] == R::e0());
res.f_[1] = constant;
LinearSieve::Omega(n_);
const auto &inverse = internal::dirichlet_integer_inverses<R>(int(msb_pos(n_)));
for (int i = 2; i <= n_; i++)
res.f_[i] = R::mul(inverse[LinearSieve::Omega(i)], f_[i]);
return res;
}
DirichletSeries log() const
{
static_assert(internal::dirichlet_has_inv<R>::value, "R::inv is required");
if (n_ == 0) return DirichletSeries(n_);
assert(f_[1] == R::e1());
if (!multiplicative_)
return (diff() / *this).integ();
DirichletSeries res(n_);
LinearSieve::reserve(n_);
const auto integers = internal::dirichlet_integers<R>(int(msb_pos(n_)));
const auto &inverse = internal::dirichlet_integer_inverses<R>(int(msb_pos(n_)));
for (int i = 2; i <= n_; i++)
{
const auto q = LinearSieve::lpf_[i];
if (i != q.pe) continue;
S value = R::mul(integers[q.e], f_[i]);
for (int a = q.p, b = i / q.p, j = 1; b > 1; a *= q.p, b /= q.p, j++)
value = R::add(value, R::minus(R::mul(R::mul(integers[j], res.f_[a]), f_[b])));
res.f_[i] = R::mul(value, inverse[q.e]);
}
return res;
}
DirichletSeries exp() const
{
static_assert(internal::dirichlet_has_inv<R>::value, "R::inv is required");
if (n_ == 0) return unit(n_);
assert(f_[1] == R::e0());
LinearSieve::Omega(n_);
const auto integers = internal::dirichlet_integers<R>(int(msb_pos(n_)));
const auto &inverse = internal::dirichlet_integer_inverses<R>(int(msb_pos(n_)));
bool primepowers_only = true;
for (int i = 2; i <= n_; i++)
if (f_[i] != R::e0() && LinearSieve::lpf_[i].pe != i)
{
primepowers_only = false;
break;
}
DirichletSeries res = unit(n_);
if (primepowers_only)
{
for (int i = 2; i <= n_; i++)
{
const auto q = LinearSieve::lpf_[i];
if (i != q.pe)
res.f_[i] = R::mul(res.f_[q.pe], res.f_[i / q.pe]);
else
{
S value = R::e0();
for (int a = q.p, b = i / q.p, j = 1;; a *= q.p, b /= q.p, j++)
{
value = R::add(value, R::mul(R::mul(integers[j], f_[a]), res.f_[b]));
if (b == 1) break;
}
res.f_[i] = R::mul(value, inverse[q.e]);
}
}
}
else
{
const auto weighted = diff();
for (int i = 1; i <= n_; i++)
{
if (i > 1) res.f_[i] = R::mul(res.f_[i], inverse[LinearSieve::Omega(i)]);
if (res.f_[i] == R::e0()) continue;
for (int j = 2; j <= n_ / i; j++)
res.f_[i * j] = R::add(res.f_[i * j], R::mul(res.f_[i], weighted.f_[j]));
}
res.multiplicative_ = false;
}
return res;
}
template <class T, enable_if_t<is_same_v<T, S> && !is_integral_ext<T>, int> = 0>
DirichletSeries pow(const T &exponent) const
{
static_assert(internal::dirichlet_has_inv<R>::value, "R::inv is required");
if (n_ == 0) return unit(n_);
assert(f_[1] == R::e1());
if (exponent == R::e0()) return unit(n_);
if (exponent == R::e1()) return *this;
return (log() * exponent).exp();
}
DirichletSeries pow(ll exponent) const
{
assert(exponent >= 0);
if (exponent == 0) return unit(n_);
if (exponent == 1) return *this;
if constexpr (internal::dirichlet_modint_field<R>::value)
{
if (n_ >= 1 && f_[1] == R::e1())
{
bool invertible = true;
for (int i = 2; i <= int(msb_pos(n_)); i++)
if (S::mod() % i == 0) { invertible = false; break; }
if (invertible)
return pow(S(exponent));
}
}
DirichletSeries res = unit(n_), base(*this);
while (exponent > 0)
{
if (exponent & 1)
res *= base;
exponent >>= 1;
if (exponent > 0)
base *= base;
}
return res;
}
};
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;
int size() const { return k + large; }
int index(ll v) const
{
if (v <= k) return int(v) - 1;
if (v > maximum) return size();
return size() - raw_index[size_t(n / ipow<ll>(v, D))];
}
};
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))];
}
S coefficient(int i) const
{
assert(1 <= i && i <= k_);
return R::add(small_[i], R::minus(small_[i - 1]));
}
public:
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); }
int size() const { return k_ + l_; }
int index(ll x) const
{
assert(x >= 1 && contains(x));
return x <= k_ ? int(x) - 1 : size() - large_index(x);
}
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)];
}
private:
void check_shape(const DirichletPrefixSum &g) const
{ assert(n_ == g.n_); (void)g; }
public:
friend DirichletPrefixSum operator*(const S &a, const DirichletPrefixSum &f) { return f * a; }
private:
public:
DirichletPrefixSum operator/(const DirichletPrefixSum &g) const
{
check_shape(g);
DirichletPrefixSum res(*this, same_shape{});
res.multiplicative_ = multiplicative_ && g.multiplicative_;
if (n_ == 0)
return res;
const internal::dirichlet_divisor<R> divide(g.small_[1]);
const auto &coordinates = [&]() -> decltype(auto)
{
if constexpr (D == 1) return internal::dirichlet_prefix_sum_index(n_);
else return (coordinates_);
}();
const int count = size();
const auto sub = [](const S &a, const S &b) { return R::add(a, R::minus(b)); };
auto at = [&](const DirichletPrefixSum &a, int i) -> const S &
{ return i <= k_ ? a.small_[i] : a.large_[count + 1 - i]; };
vc<S> delta(size_t(count) + 2, R::e0());
for (int i = 1; i <= count; i++) delta[i] = sub(at(*this, i), at(*this, i - 1));
res.small_[1] = divide(small_[1]);
const S h1 = res.small_[1];
auto subtract = [&](int lo, int hi, const S &term)
{
delta[lo] = sub(delta[lo], term);
delta[hi + 1] = R::add(delta[hi + 1], term);
};
for (int i = 2; i <= count; i++)
{
const ll x = value(i - 1);
using Index = conditional_t<D == 1, int, ll>;
const Index z = [&]() -> Index
{
if constexpr (D == 1) return count + 1 - i;
else return n_ / ipow<ll>(x, D);
}();
S gathered = R::e0();
for (int u = 2; ; u++)
{
const Index lo = max<Index>(u, z);
const ll hi = x / u;
if (hi <= lo) break;
gathered = R::add(gathered, R::add(
R::mul(res.coefficient(u), sub(g.F(hi), g.small_[lo])),
R::mul(g.coefficient(u), sub(res.F(hi), res.small_[lo]))));
}
subtract(i, i, gathered);
const S block = divide(sub(delta[i], R::mul(h1, sub(at(g, i), at(g, i - 1)))));
(i <= k_ ? res.small_[i] : res.large_[count + 1 - i]) = R::add(at(res, i - 1), block);
if (i > k_) continue;
const S hi = res.coefficient(i), gi = g.coefficient(i);
const ll bound = iroot(n_ / i, D);
const int end = coordinates.index(bound) + 1;
const int limit = int(min<ll>(i - 1, bound / i));
for (int v = 2; v <= limit; v++)
{
const S term = R::add(R::mul(hi, g.coefficient(v)), R::mul(gi, res.coefficient(v)));
subtract(coordinates.index(ll(i) * v) + 1, end, term);
}
const int diagonal = coordinates.index(ll(i) * i) + 1;
if constexpr (D == 1) subtract(diagonal, count, R::mul(hi, gi));
else if (diagonal <= count) subtract(diagonal, count, R::mul(hi, gi));
}
return res;
}
DirichletPrefixSum inv() const { return unit(n_) / *this; }
DirichletPrefixSum pow(ll exponent) const
{
assert(exponent >= 0);
DirichletPrefixSum res = unit(n_), base(*this);
while (exponent > 0)
{
if (exponent & 1)
res *= base;
exponent >>= 1;
if (exponent > 0)
base *= base;
}
return res;
}
};
// https://github.com/miscalculation53/library/tree/wip/ds/fenwick_tree/fenwick_tree.hpp
namespace internal
{
inline constexpr ll multiplicative_prefix_sum_fenwick_threshold = 1'000'000;
}
void main2()
{
LL(N);
DirichletPrefixSum<RingAddSubMul<ll>> zeta(N, zeta_prefix_sum);
auto mu = zeta.inv();
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() {}