結果
| 問題 |
No.2266 Fractions (hard)
|
| ユーザー |
 NyaanNyaan
|
| 提出日時 | 2023-04-07 23:53:15 |
| 言語 | C++17(gcc12) (gcc 12.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 5,669 ms / 6,000 ms |
| コード長 | 31,677 bytes |
| コンパイル時間 | 13,504 ms |
| コンパイル使用メモリ | 310,356 KB |
| 最終ジャッジ日時 | 2025-02-12 03:18:40 |
|
ジャッジサーバーID （参考情報） |
judge3 / judge5 |
| 純コード判定しない問題か言語 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 35 |
ソースコード
/**
* date : 2023-04-07 23:53:12
*/
#define NDEBUG
using namespace std;
// intrinstic
#include <immintrin.h>
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
// utility
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T, typename U>
struct P : pair<T, U> {
template <typename... Args>
P(Args... args) : pair<T, U>(args...) {}
using pair<T, U>::first;
using pair<T, U>::second;
P &operator+=(const P &r) {
first += r.first;
second += r.second;
return *this;
}
P &operator-=(const P &r) {
first -= r.first;
second -= r.second;
return *this;
}
P &operator*=(const P &r) {
first *= r.first;
second *= r.second;
return *this;
}
template <typename S>
P &operator*=(const S &r) {
first *= r, second *= r;
return *this;
}
P operator+(const P &r) const { return P(*this) += r; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator*(const P &r) const { return P(*this) *= r; }
template <typename S>
P operator*(const S &r) const {
return P(*this) *= r;
}
P operator-() const { return P{-first, -second}; }
};
using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;
template <typename T>
int sz(const T &t) {
return t.size();
}
template <typename T, typename U>
inline bool amin(T &x, U y) {
return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
inline T Max(const vector<T> &v) {
return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
return accumulate(begin(v), end(v), 0LL);
}
template <typename T>
int lb(const vector<T> &v, const T &a) {
return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
return upper_bound(begin(v), end(v), a) - begin(v);
}
constexpr long long TEN(int n) {
long long ret = 1, x = 10;
for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
return ret;
}
template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
return make_pair(t, u);
}
template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
vector<T> ret(v.size() + 1);
if (rev) {
for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
} else {
for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
}
return ret;
};
template <typename T>
vector<T> mkuni(const vector<T> &v) {
vector<T> ret(v);
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());
return ret;
}
template <typename F>
vector<int> mkord(int N,F f) {
vector<int> ord(N);
iota(begin(ord), end(ord), 0);
sort(begin(ord), end(ord), f);
return ord;
}
template <typename T>
vector<int> mkinv(vector<T> &v) {
int max_val = *max_element(begin(v), end(v));
vector<int> inv(max_val + 1, -1);
for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
return inv;
}
vector<int> mkiota(int n) {
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
return ret;
}
template <typename T>
T mkrev(const T &v) {
T w{v};
reverse(begin(w), end(w));
return w;
}
template <typename T>
bool nxp(vector<T> &v) {
return next_permutation(begin(v), end(v));
}
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
} // namespace Nyaan
// bit operation
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} // namespace Nyaan
// inout
namespace Nyaan {
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << p.first << " " << p.second;
return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
int s = (int)v.size();
for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
for (auto &x : v) is >> x;
return is;
}
istream &operator>>(istream &is, __int128_t &x) {
string S;
is >> S;
x = 0;
int flag = 0;
for (auto &c : S) {
if (c == '-') {
flag = true;
continue;
}
x *= 10;
x += c - '0';
}
if (flag) x = -x;
return is;
}
istream &operator>>(istream &is, __uint128_t &x) {
string S;
is >> S;
x = 0;
for (auto &c : S) {
x *= 10;
x += c - '0';
}
return is;
}
ostream &operator<<(ostream &os, __int128_t x) {
if (x == 0) return os << 0;
if (x < 0) os << '-', x = -x;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
if (x == 0) return os << 0;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
struct IoSetupNya {
IoSetupNya() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetupnya;
} // namespace Nyaan
// debug
#ifdef NyaanDebug
#define trc(...) (void(0))
#else
#define trc(...) (void(0))
#endif
#ifdef NyaanLocal
#define trc2(...) (void(0))
#else
#define trc2(...) (void(0))
#endif
// macro
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
int __VA_ARGS__; \
in(__VA_ARGS__)
#define inl(...) \
long long __VA_ARGS__; \
in(__VA_ARGS__)
#define ins(...) \
string __VA_ARGS__; \
in(__VA_ARGS__)
#define in2(s, t) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i]); \
}
#define in3(s, t, u) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i]); \
}
#define in4(s, t, u, v) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i], v[i]); \
}
#define die(...) \
do { \
Nyaan::out(__VA_ARGS__); \
return; \
} while (0)
namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }
//
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
unsigned int _m;
unsigned long long im;
// @param m `1 <= m < 2^31`
barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if (m0 < 0) m0 += b / s;
return {s, m0};
}
// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
} // namespace internal
} // namespace atcoder
namespace atcoder {
long long pow_mod(long long x, long long n, int m) {
assert(0 <= n && 1 <= m);
if (m == 1) return 0;
internal::barrett bt((unsigned int)(m));
unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
while (n) {
if (n & 1) r = bt.mul(r, y);
y = bt.mul(y, y);
n >>= 1;
}
return r;
}
long long inv_mod(long long x, long long m) {
assert(1 <= m);
auto z = internal::inv_gcd(x, m);
assert(z.first == 1);
return z.second;
}
// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long>& r,
const std::vector<long long>& m) {
assert(r.size() == m.size());
int n = int(r.size());
// Contracts: 0 <= r0 < m0
long long r0 = 0, m0 = 1;
for (int i = 0; i < n; i++) {
assert(1 <= m[i]);
long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
if (m0 < m1) {
std::swap(r0, r1);
std::swap(m0, m1);
}
if (m0 % m1 == 0) {
if (r0 % m1 != r1) return {0, 0};
continue;
}
// assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)
// (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
// r2 % m0 = r0
// r2 % m1 = r1
// -> (r0 + x*m0) % m1 = r1
// -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
// -> x = (r1 - r0) / g * inv(u0) (mod u1)
// im = inv(u0) (mod u1) (0 <= im < u1)
long long g, im;
std::tie(g, im) = internal::inv_gcd(m0, m1);
long long u1 = (m1 / g);
// |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
if ((r1 - r0) % g) return {0, 0};
// u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
long long x = (r1 - r0) / g % u1 * im % u1;
// |r0| + |m0 * x|
// < m0 + m0 * (u1 - 1)
// = m0 + m0 * m1 / g - m0
// = lcm(m0, m1)
r0 += x * m0;
m0 *= u1; // -> lcm(m0, m1)
if (r0 < 0) r0 += m0;
}
return {r0, m0};
}
long long floor_sum(long long n, long long m, long long a, long long b) {
long long ans = 0;
if (a >= m) {
ans += (n - 1) * n * (a / m) / 2;
a %= m;
}
if (b >= m) {
ans += n * (b / m);
b %= m;
}
long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
if (y_max == 0) return ans;
ans += (n - (x_max + a - 1) / a) * y_max;
ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
return ans;
}
} // namespace atcoder
// { (q, l, r) : forall x in (l,r], floor(N/x) = q }
// を引数に取る関数f(q, l, r)を渡す。範囲が左に半開なのに注意
template <typename T, typename F>
void enumerate_quotient(T N, const F& f) {
T sq = sqrt(N), upper = N, quo = 0;
while (upper > sq) {
T thres = N / (++quo + 1);
f(quo, thres, upper);
upper = thres;
}
while (upper > 0) {
f(N / upper, upper - 1, upper);
upper--;
}
}
/**
* @brief 商の列挙
*/
struct Rational {
using R = Rational;
using i128 = __int128_t;
using i64 = long long;
using u64 = unsigned long long;
long long x, y;
Rational() : x(0), y(1) {}
Rational(long long _x, long long _y = 1) : x(_x), y(_y) {
assert(y != 0);
if (_y != 1) {
long long g = gcd(x, y);
if (g != 0) x /= g, y /= g;
if (y < 0) x = -x, y = -y;
}
}
u64 gcd(i64 A, i64 B) {
u64 a = A >= 0 ? A : -A;
u64 b = B >= 0 ? B : -B;
if (a == 0 || b == 0) return a + b;
int n = __builtin_ctzll(a);
int m = __builtin_ctzll(b);
a >>= n;
b >>= m;
while (a != b) {
int d = __builtin_ctzll(a - b);
bool f = a > b;
u64 c = f ? a : b;
b = f ? b : a;
a = (c - b) >> d;
}
return a << min(n, m);
}
friend R operator+(const R& l, const R& r) {
return R(l.x * r.y + l.y * r.x, l.y * r.y);
}
friend R operator-(const R& l, const R& r) {
return R(l.x * r.y - l.y * r.x, l.y * r.y);
}
friend R operator*(const R& l, const R& r) { return R(l.x * r.x, l.y * r.y); }
friend R operator/(const R& l, const R& r) {
assert(r.x != 0);
return R(l.x * r.y, l.y * r.x);
}
R& operator+=(const R& r) { return (*this) = (*this) + r; }
R& operator-=(const R& r) { return (*this) = (*this) - r; }
R& operator*=(const R& r) { return (*this) = (*this) * r; }
R& operator/=(const R& r) { return (*this) = (*this) / r; }
R operator-() const {
R r;
r.x = -x, r.y = y;
return r;
}
R inverse() const {
assert(x != 0);
R r;
r.x = y, r.y = x;
if (x < 0) r.x = -r.x, r.y = -r.y;
return r;
}
R pow(long long p) const {
R res(1), base(*this);
while (p) {
if (p & 1) res *= base;
base *= base;
p >>= 1;
}
return res;
}
friend bool operator==(const R& l, const R& r) {
return l.x == r.x && l.y == r.y;
};
friend bool operator!=(const R& l, const R& r) {
return l.x != r.x || l.y != r.y;
};
friend bool operator<(const R& l, const R& r) {
return i128(l.x) * r.y < i128(l.y) * r.x;
};
friend bool operator<=(const R& l, const R& r) { return l < r || l == r; }
friend bool operator>(const R& l, const R& r) {
return i128(l.x) * r.y > i128(l.y) * r.x;
};
friend bool operator>=(const R& l, const R& r) { return l > r || l == r; }
friend ostream& operator<<(ostream& os, const R& r) {
os << r.x;
if (r.x != 0 && r.y != 1) os << "/" << r.y;
return os;
}
long long toMint(long long mod) {
assert(mod != 0);
i64 a = y, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b);
swap(u -= t * v, v);
}
return i128((u % mod + mod) % mod) * x % mod;
}
};
template <typename R = Rational>
struct Binomial {
vector<R> fc;
Binomial(int = 0) { fc.emplace_back(1); }
void extend() {
int n = fc.size();
R nxt = fc.back() * n;
fc.push_back(nxt);
}
R fac(int n) {
while ((int)fc.size() <= n) extend();
return fc[n];
}
R finv(int n) { return fac(n).inverse(); }
R inv(int n) { return R{1, max(n, 1)}; }
R C(int n, int r) {
if (n < 0 or r < 0 or n < r) return R{0};
return fac(n) * finv(n - r) * finv(r);
}
R operator()(int n, int r) { return C(n, r); }
template <typename I>
R multinomial(const vector<I>& r) {
static_assert(is_integral<I>::value == true);
int n = 0;
for (auto& x : r) {
if (x < 0) return R{0};
n += x;
}
R res = fac(n);
for (auto& x : r) res *= finv(x);
return res;
}
template <typename I>
R operator()(const vector<I>& r) {
return multinomial(r);
}
};
// Prime Sieve {2, 3, 5, 7, 11, 13, 17, ...}
vector<int> prime_enumerate(int N) {
vector<bool> sieve(N / 3 + 1, 1);
for (int p = 5, d = 4, i = 1, sqn = sqrt(N); p <= sqn; p += d = 6 - d, i++) {
if (!sieve[i]) continue;
for (int q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p,
qe = sieve.size();
q < qe; q += r = s - r)
sieve[q] = 0;
}
vector<int> ret{2, 3};
for (int p = 5, d = 4, i = 1; p <= N; p += d = 6 - d, i++)
if (sieve[i]) ret.push_back(p);
while (!ret.empty() && ret.back() > N) ret.pop_back();
return ret;
}
struct divisor_transform {
template <typename T>
static void zeta_transform(vector<T> &a) {
int N = a.size() - 1;
auto sieve = prime_enumerate(N);
for (auto &p : sieve)
for (int k = 1; k * p <= N; ++k) a[k * p] += a[k];
}
template <typename T>
static void mobius_transform(T &a) {
int N = a.size() - 1;
auto sieve = prime_enumerate(N);
for (auto &p : sieve)
for (int k = N / p; k > 0; --k) a[k * p] -= a[k];
}
template <typename I, typename T>
static void zeta_transform(map<I, T> &a) {
for (auto p = rbegin(a); p != rend(a); p++)
for (auto &x : a) {
if (p->first == x.first) break;
if (p->first % x.first == 0) p->second += x.second;
}
}
template <typename I, typename T>
static void mobius_transform(map<I, T> &a) {
for (auto &x : a) {
for (auto p = rbegin(a); p != rend(a); p++) {
if (x.first == p->first) break;
if (p->first % x.first == 0) p->second -= x.second;
}
}
}
};
struct multiple_transform {
template <typename T>
static void zeta_transform(vector<T> &a) {
int N = a.size() - 1;
auto sieve = prime_enumerate(N);
for (auto &p : sieve)
for (int k = N / p; k > 0; --k) a[k] += a[k * p];
}
template <typename T>
static void mobius_transform(vector<T> &a) {
int N = a.size() - 1;
auto sieve = prime_enumerate(N);
for (auto &p : sieve)
for (int k = 1; k * p <= N; ++k) a[k] -= a[k * p];
}
template <typename I, typename T>
static void zeta_transform(map<I, T> &a) {
for (auto &x : a)
for (auto p = rbegin(a); p->first != x.first; p++)
if (p->first % x.first == 0) x.second += p->second;
}
template <typename I, typename T>
static void mobius_transform(map<I, T> &a) {
for (auto p1 = rbegin(a); p1 != rend(a); p1++)
for (auto p2 = rbegin(a); p2 != p1; p2++)
if (p2->first % p1->first == 0) p1->second -= p2->second;
}
};
/**
* @brief 倍数変換・約数変換
* @docs docs/multiplicative-function/divisor-multiple-transform.md
*/
// f(p, c) : f(p^c) の値を返す
template <typename T, T (*f)(long long, long long)>
struct mf_prefix_sum {
using i64 = long long;
i64 M, sq, s;
vector<int> p;
int ps;
vector<T> buf;
T ans;
mf_prefix_sum(i64 m) : M(m) {
assert(m < (1LL << 42));
sq = sqrt(M);
while (sq * sq > M) sq--;
while ((sq + 1) * (sq + 1) <= M) sq++;
if (M != 0) {
i64 hls = md(M, sq);
if (hls != 1 && md(M, hls - 1) == sq) hls--;
s = hls + sq;
p = prime_enumerate(sq);
ps = p.size();
ans = T{};
}
}
// 素数の個数関数に関するテーブル
vector<T> pi_table() {
if (M == 0) return {};
i64 hls = md(M, sq);
if (hls != 1 && md(M, hls - 1) == sq) hls--;
vector<i64> hl(hls);
for (int i = 1; i < hls; i++) hl[i] = md(M, i) - 1;
vector<int> hs(sq + 1);
iota(begin(hs), end(hs), -1);
int pi = 0;
for (auto& x : p) {
i64 x2 = i64(x) * x;
i64 imax = min<i64>(hls, md(M, x2) + 1);
for (i64 i = 1, ix = x; i < imax; ++i, ix += x) {
hl[i] -= (ix < hls ? hl[ix] : hs[md(M, ix)]) - pi;
}
for (int n = sq; n >= x2; n--) hs[n] -= hs[md(n, x)] - pi;
pi++;
}
vector<T> res;
res.reserve(2 * sq + 10);
for (auto& x : hl) res.push_back(x);
for (int i = hs.size(); --i;) res.push_back(hs[i]);
assert((int)res.size() == s);
return res;
}
// 素数の prefix sum に関するテーブル
vector<T> prime_sum_table() {
if (M == 0) return {};
i64 hls = md(M, sq);
if (hls != 1 && md(M, hls - 1) == sq) hls--;
vector<T> h(s);
T inv2 = T{2}.inverse();
for (int i = 1; i < hls; i++) {
T x = md(M, i);
h[i] = x * (x + 1) * inv2 - 1;
}
for (int i = 1; i <= sq; i++) {
T x = i;
h[s - i] = x * (x + 1) / 2 - 1;
}
for (auto& x : p) {
T xt = x;
T pi = h[s - x + 1];
i64 x2 = i64(x) * x;
i64 imax = min<i64>(hls, md(M, x2) + 1);
i64 ix = x;
for (i64 i = 1; i < imax; ++i, ix += x) {
h[i] -= ((ix < hls ? h[ix] : h[s - md(M, ix)]) - pi) * xt;
}
for (int n = sq; n >= x2; n--) {
h[s - n] -= (h[s - md(n, x)] - pi) * xt;
}
}
assert((int)h.size() == s);
return h;
}
void dfs(int i, int c, i64 prod, T cur) {
ans += cur * f(p[i], c + 1);
i64 lim = md(M, prod);
if (lim >= 1LL * p[i] * p[i]) dfs(i, c + 1, p[i] * prod, cur);
cur *= f(p[i], c);
ans += cur * (buf[idx(lim)] - buf[idx(p[i])]);
int j = i + 1;
// M < 2**42 -> p_j < 2**21 -> (p_j)^3 < 2**63
for (; j < ps && 1LL * p[j] * p[j] * p[j] <= lim; j++) {
dfs(j, 1, prod * p[j], cur);
}
for (; j < ps && 1LL * p[j] * p[j] <= lim; j++) {
T sm = f(p[j], 2);
int id1 = idx(md(lim, p[j])), id2 = idx(p[j]);
sm += f(p[j], 1) * (buf[id1] - buf[id2]);
ans += cur * sm;
}
}
// fprime 破壊的
T run(vector<T>& fprime) {
if (M == 0) return {};
set_buf(fprime);
assert((int)buf.size() == s);
ans = buf[idx(M)] + 1;
for (int i = 0; i < ps; i++) dfs(i, 1, p[i], 1);
return ans;
}
i64 md(i64 n, i64 d) { return double(n) / d; }
i64 idx(i64 n) { return n <= sq ? s - n : md(M, n); }
void set_buf(vector<T>& _buf) { swap(buf, _buf); }
};
/**
* @brief 乗法的関数のprefix sum
* @docs docs/multiplicative-function/sum-of-multiplicative-function.md
*/
// f(n, p, c) : n = pow(p, c), f is multiplicative function
template <typename T, T (*f)(int, int, int)>
struct enamurate_multiplicative_function {
enamurate_multiplicative_function(int _n)
: ps(prime_enumerate(_n)), a(_n + 1, T()), n(_n), p(ps.size()) {}
vector<T> run() {
a[1] = 1;
dfs(-1, 1, 1);
return a;
}
private:
vector<int> ps;
vector<T> a;
int n, p;
void dfs(int i, long long x, T y) {
a[x] = y;
if (y == T()) return;
for (int j = i + 1; j < p; j++) {
long long nx = x * ps[j];
long long dx = ps[j];
if (nx > n) break;
for (int c = 1; nx <= n; nx *= ps[j], dx *= ps[j], ++c) {
dfs(j, nx, y * f(dx, ps[j], c));
}
}
}
};
/**
* @brief 乗法的関数の列挙
*/
namespace multiplicative_function {
template <typename T>
T moebius(int, int, int c) {
return c == 0 ? 1 : c == 1 ? -1 : 0;
}
template <typename T>
T sigma0(int, int, int c) {
return c + 1;
}
template <typename T>
T sigma1(int n, int p, int) {
return (n - 1) / (p - 1) + n;
}
template <typename T>
T totient(int n, int p, int) {
return n - n / p;
}
} // namespace multiplicative_function
template <typename T>
static constexpr vector<T> mobius_function(int n) {
enamurate_multiplicative_function<T, multiplicative_function::moebius<T>> em(
n);
return em.run();
}
template <typename T>
static constexpr vector<T> sigma0(int n) {
enamurate_multiplicative_function<T, multiplicative_function::sigma0<T>> em(
n);
return em.run();
}
template <typename T>
static constexpr vector<T> sigma1(int n) {
enamurate_multiplicative_function<T, multiplicative_function::sigma1<T>> em(
n);
return em.run();
}
template <typename T>
static constexpr vector<T> totient(int n) {
enamurate_multiplicative_function<T, multiplicative_function::totient<T>> em(
n);
return em.run();
}
/**
* @brief 有名な乗法的関数
* @docs docs/multiplicative-function/mf-famous-series.md
*/
template <typename T>
T sum_of_totient(long long N) {
if (N < 2) return N;
using i64 = long long;
auto f = [](i64 v, i64 p, i64) -> i64 { return v / p * (p - 1); };
vector<i64> ns{0}, p;
for (i64 i = N; i > 0; i = N / (N / i + 1)) ns.push_back(i);
i64 s = ns.size(), sq = sqrt(N);
auto idx = [&](i64 n) { return n <= sq ? s - n : N / n; };
vector<T> h0(s), h1(s), buf(s);
for (int i = 0; i < s; i++) {
T x = ns[i];
h0[i] = x - 1;
h1[i] = x * (x + 1) / 2 - 1;
}
for (i64 x = 2; x <= sq; ++x) {
if (h0[s - x] == h0[s - x + 1]) continue;
p.push_back(x);
i64 x2 = x * x;
for (i64 i = 1, n = ns[i]; i < s && n >= x2; n = ns[++i]) {
int id = (i * x <= sq ? i * x : s - n / x);
h0[i] -= h0[id] - h0[s - x + 1];
h1[i] -= (h1[id] - h1[s - x + 1]) * x;
}
}
for (int i = 0; i < s; i++) buf[i] = h1[i] - h0[i];
T ans = buf[idx(N)] + 1;
auto dfs = [&](auto rec, int i, int c, i64 v, i64 lim, T cur) -> void {
ans += cur * f(p[i] * v, p[i], c + 1);
if (lim >= p[i] * p[i]) rec(rec, i, c + 1, p[i] * v, lim / p[i], cur);
cur *= f(v, p[i], c);
ans += cur * (buf[idx(lim)] - buf[idx(p[i])]);
for (int j = i + 1; j < (int)p.size() && p[j] * p[j] <= lim; j++) {
rec(rec, j, 1, p[j], lim / p[j], cur);
}
};
for (int i = 0; i < (int)p.size(); i++) dfs(dfs, i, 1, p[i], N / p[i], 1);
return ans;
}
/**
* @brief トーシェント関数の和
*/
using namespace Nyaan;
V<short> mo;
void precalc(ll) {
if (mo.empty()) {
mo = mobius_function<short>(TEN(8) + 200);
rep(i, sz(mo) - 1) mo[i + 1] += mo[i];
}
}
// k 番目に小さい
pl calc(ll N, ll K) {
precalc(N);
auto cnt = [&](Rational f) -> ll {
ll s = 0;
enumerate_quotient(N, [&](ll q, ll l, ll r) {
ll x = 0;
x += atcoder::floor_sum(r + 1, f.y, f.x, 0);
x -= atcoder::floor_sum(l + 1, f.y, f.x, 0);
s += x * mo[q];
});
/*
each(i, mop) {
if (i > N) break;
s += i * ll(f.x) / int(f.y);
}
*/
trc2(f, s);
return s;
};
Rational L{0, 1};
Rational M{1, 2};
Rational R{1, 1};
while (true) {
// trc2(L.x, L.y, M.x, M.y, R.x, R.y);
ll c = cnt(M);
if (c == K) {
break;
}
if (c < K) {
for (ll i = 1;; i *= 2) {
Rational f{L.x + R.x * i, L.y + R.y * i};
if (max(f.x, f.y) > N) break;
if (cnt(f) == K) return {f.x, f.y};
if (cnt(f) < K) {
L = f;
} else {
break;
}
}
} else {
for (ll i = 1;; i *= 2) {
Rational f{L.x * i + R.x, L.y * i + R.y};
if (max(f.x, f.y) > N) break;
if (cnt(f) == K) return {f.x, f.y};
if (cnt(f) >= K) {
R = f;
} else {
break;
}
}
}
M = Rational{L.x + R.x, L.y + R.y};
}
return {M.x, M.y};
}
void q() {
inl(N, K);
ll s = sum_of_totient<ll>(N) - 1;
trc(s);
ll p = -1, q = -1;
if (K <= s) {
tie(p, q) = calc(N, K);
} else if (K == s + 1) {
p = q = 1;
} else if (K <= s * 2 + 1) {
tie(q, p) = calc(N, 2 * s + 1 - (K - 1));
} else {
// do nothing
}
if (p == -1) {
out(-1);
} else {
cout << p << "/" << q << "\n";
}
}
void Nyaan::solve() {
int t = 1;
// in(t);
while (t--) q();
}