#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else #pragma GCC optimize("Ofast") #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) \ for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template T ceil(T x, T y) { return floor(x + y - 1, y); } template T bmod(T x, T y) { return x - y * floor(x, y); } template pair divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { T a = que.top(); que.pop(); return a; } template T POP(vc &que) { T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io.hpp" #define FASTIO #include // https://judge.yosupo.jp/submission/21623 namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf 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, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(u32 &x) { rd_integer(x); } void rd(u64 &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } void rd(f128 &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d: x) rd(d); } template void rd(vc &x) { for (auto &d: x) rd(d); } void read() {} template void read(H &h, T &... t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c: s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(u32 x) { wt_integer(x); } void wt(u64 x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } void wt(f128 x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ read(__VA_ARGS__) #define U64(...) \ u64 __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 3 "main.cpp" #line 1 "/home/maspy/compro/library/nt/array_on_floor.hpp" // N=10 だと dat = {dp[1], dp[2], dp[3], dp[5], dp[10]} みたいになる // hashmap より数倍高速 template struct Array_On_Floor { u64 N; u32 n, sq; vc dat; Array_On_Floor() {} Array_On_Floor(u64 N, T default_value = T{}) : N(N) { assert(N <= u64(1) << 50); sq = sqrtl(N); n = (u64(sq) * sq + sq <= N ? sq : sq - 1); dat.resize(n + sq, default_value); } u32 size() { return dat.size(); } T& operator[](u64 d) { int i = get_index(d); return dat[i]; } inline u32 get_index(u64 d) { assert(d > 0); if (d <= n) return d - 1; return dat.size() - u32(double(N) / d); } // dat[i] に対応する floor u64 get_floor(u32 i) { return (i < n ? 1 + i : double(N) / (n + sq - i)); } template void enumerate_all(F f) { FOR(i, len(dat)) { f(get_floor(i), dat[i]); } } }; #line 2 "/home/maspy/compro/library/nt/primetable.hpp" template vc primetable(int LIM) { ++LIM; const int S = 32768; static int done = 2; static vc primes = {2}, sieve(S + 1); if (done < LIM) { done = LIM; primes = {2}, sieve.assign(S + 1, 0); const int R = LIM / 2; primes.reserve(int(LIM / log(LIM) * 1.1)); vc> cp; for (int i = 3; i <= S; i += 2) { if (!sieve[i]) { cp.eb(i, i * i / 2); for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1; } } for (int L = 1; L <= R; L += S) { array block{}; for (auto& [p, idx]: cp) for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1; FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1); } } int k = LB(primes, LIM + 1); return {primes.begin(), primes.begin() + k}; } #line 3 "/home/maspy/compro/library/nt/zeta.hpp" template void divisor_zeta(vc& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3(x, 1, N / p + 1) A[p * x] += A[x]; } } template void divisor_mobius(vc& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3_R(x, 1, N / p + 1) A[p * x] -= A[x]; } } template void multiplier_zeta(vc& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3_R(x, 1, N / p + 1) A[x] += A[p * x]; } } template void multiplier_mobius(vc& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3(x, 1, N / p + 1) A[x] -= A[p * x]; } } #line 2 "/home/maspy/compro/library/nt/mobius_table.hpp" template vc mobius_table(int N){ vc mu(N + 1); mu[1] = T(1); divisor_mobius(mu); return mu; } #line 1 "/home/maspy/compro/library/enumerate/floor_range.hpp" // 商が q の区間 [l,r) を q について昇順 template void floor_range(u64 N, F f) { assert(N <= (u64(1) << 50)); u64 sq = sqrtl(N); u32 n = (sq * sq + sq <= N ? sq : sq - 1); u64 prev = N + 1; for (u32 q = 1; q <= n; ++q) { u64 x = double(N) / (q + 1) + 1; f(q, x, prev), prev = x; } for (u32 l = sq; l >= 1; --l) { f(u64(double(N) / l), l, l + 1); } } #line 4 "/home/maspy/compro/library/nt/mertens.hpp" template struct Mertens { Array_On_Floor sum; Mertens() {} Mertens(u64 N, u64 K = -1) { build(N, K); } void build(u64 N, u64 K = -1) { sum = Array_On_Floor(N); if (K == u64(-1)) { K = pow(N, 0.67); } vc A = mobius_table(K); FOR(k, 1, K) A[k + 1] += A[k]; FOR(i, len(sum)) { u64 n = sum.get_floor(i); if (n <= K) { sum.dat[i] = A[n]; continue; } T ans = 1; floor_range(n, [&](u64 q, u64 l, u64 r) -> void { if (q == n) return; ans -= sum[q] * T(r - l); }); sum.dat[i] = ans; } } T operator[](u64 n) { return sum[n]; } }; #line 2 "/home/maspy/compro/library/mod/floor_sum_of_linear.hpp" // sum_{x in [L,R)} floor(ax + b, mod) // I は範囲内で ax+b がオーバーフローしない程度 template O floor_sum_of_linear(I L, I R, I a, I b, I mod) { assert(L <= R); O res = 0; b += L * a; I N = R - L; if (b < 0) { I k = ceil(-b, mod); b += k * mod; res -= O(N) * O(k); } while (N) { I q; tie(q, a) = divmod(a, mod); res += (N & 1 ? O(N) * O((N - 1) / 2) * O(q) : O(N / 2) * O(N - 1) * O(q)); if (b >= mod) { tie(q, b) = divmod(b, mod); res += O(N) * q; } tie(N, b) = divmod(a * N + b, mod); tie(a, mod) = mp(mod, a); } return res; } #line 6 "main.cpp" // 最大分母 N を指定する // 既約分数を数えたり k 番目を求めたりする struct Range_Rational_Count { u32 N; u64 total; Mertens M; Range_Rational_Count(u32 N) : N(N), M(N) { total = count(1, 1); } // [0, a/b) u64 count(u32 a, u32 b) { assert(a <= b); if (a == 0) return 0; // [0,a/b] u64 ans = 1; floor_range(N, [&](u32 q, u32 l, u32 r) -> void { ans += u64(M[q]) * floor_sum_of_linear(l, r, a, 0, b); }); // a/b if (b <= N) --ans; return ans; } // [0,1) の中で k 番目 pair kth(u64 k) { assert(k < total); u32 int_part = k / total; k %= total; map, u64> MP; auto query = [&](u32 a, u32 b) -> u64 { pair k = {a, b}; if (MP.count(k)) return MP[k]; return MP[k] = count(a, b); }; // k 個以下なものの max u32 a = 0, b = 1, c = 1, d = 1; while (b + d <= N) { // 右に進む u32 l = 0, r = 1; while (b + r * d <= N && query(a + r * c, b + r * d) <= k) { l = r, r = 2 * r; } while (l + 1 < r) { u32 m = l + (r - l) / 2; (query(a + m * c, b + m * d) <= k ? l : r) = m; } a += l * c, b += l * d; // 左に進む l = 0, r = 1; while (r * b + d <= N && query(r * a + c, r * b + d) > k) { l = r, r = 2 * r; } while (l + 1 < r) { u32 m = l + (r - l) / 2; (query(m * a + c, m * b + d) > k ? l : r) = m; } c += l * a, d += l * b; } return {int_part * b + a, b}; } }; void solve() { u64 N, K; read(N, K); auto out = [&](u32 a, u32 b) -> void { string s = to_string(a) + "/" + to_string(b); print(s); }; Range_Rational_Count X(N); u64 t = X.total; if (K < t) { auto [a, b] = X.kth(K); out(a, b); } elif (K == t) { out(1, 1); } elif (K < 2 * t) { auto [a, b] = X.kth(2 * t - K); out(b, a); } else { print(-1); } } signed main() { solve(); return 0; }