結果

問題 No.2262 Fractions
ユーザー 👑 NachiaNachia
提出日時 2023-04-08 01:07:19
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 181 ms / 2,000 ms
コード長 8,569 bytes
コンパイル時間 1,347 ms
コンパイル使用メモリ 96,064 KB
実行使用メモリ 7,964 KB
最終ジャッジ日時 2024-05-05 07:27:08
合計ジャッジ時間 4,281 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge4
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 12 ms
6,400 KB
testcase_01 AC 181 ms
5,376 KB
testcase_02 AC 153 ms
5,376 KB
testcase_03 AC 138 ms
5,376 KB
testcase_04 AC 140 ms
5,376 KB
testcase_05 AC 126 ms
5,376 KB
testcase_06 AC 148 ms
5,376 KB
testcase_07 AC 139 ms
5,376 KB
testcase_08 AC 120 ms
5,376 KB
testcase_09 AC 137 ms
5,376 KB
testcase_10 AC 126 ms
5,376 KB
testcase_11 AC 38 ms
5,376 KB
testcase_12 AC 40 ms
5,376 KB
testcase_13 AC 38 ms
5,376 KB
testcase_14 AC 39 ms
5,376 KB
testcase_15 AC 38 ms
5,376 KB
testcase_16 AC 9 ms
5,376 KB
testcase_17 AC 9 ms
5,376 KB
testcase_18 AC 9 ms
5,376 KB
testcase_19 AC 20 ms
6,140 KB
testcase_20 AC 17 ms
5,916 KB
testcase_21 AC 19 ms
6,192 KB
testcase_22 AC 19 ms
6,008 KB
testcase_23 AC 18 ms
5,376 KB
testcase_24 AC 17 ms
7,836 KB
testcase_25 AC 17 ms
7,832 KB
testcase_26 AC 19 ms
7,836 KB
testcase_27 AC 19 ms
7,960 KB
testcase_28 AC 18 ms
7,832 KB
testcase_29 AC 18 ms
7,832 KB
testcase_30 AC 16 ms
7,960 KB
testcase_31 AC 18 ms
7,964 KB
testcase_32 AC 18 ms
7,836 KB
testcase_33 AC 19 ms
7,960 KB
testcase_34 AC 17 ms
7,836 KB
testcase_35 AC 14 ms
7,836 KB
testcase_36 AC 13 ms
7,964 KB
testcase_37 AC 7 ms
5,632 KB
testcase_38 AC 7 ms
5,632 KB
testcase_39 AC 17 ms
5,504 KB
testcase_40 AC 18 ms
5,636 KB
testcase_41 AC 21 ms
5,632 KB
testcase_42 AC 18 ms
5,632 KB
testcase_43 AC 16 ms
5,628 KB
testcase_44 AC 19 ms
7,832 KB
testcase_45 AC 19 ms
7,836 KB
権限があれば一括ダウンロードができます

ソースコード

diff # 

#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"

#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\prime-sieve-explicit.hpp"

#include <vector>
#include <algorithm>
#include <cassert>
#include <iostream>


namespace nachia{

namespace prime_sieve_explicit_internal{
    std::vector<bool> isprime = { false }; // a[x] := isprime(2x+1)

    void CalcIsPrime(int z){
        if((int)isprime.size() *2+1 < z+1){
            int new_z = isprime.size();
            while(new_z*2+1 < z+1) new_z *= 2;
            z = new_z-1;
            isprime.resize(z+1, true);
            for(int i=1; i*(i+1)*2<=z; i++) if(isprime[i]){
                for(int j=i*(i+1)*2; j<=z; j+=i*2+1) isprime[j] = false;
            }
        }
    }
    
    std::vector<int> prime_list = {2};
    int prime_list_max = 0;

    void CalcPrimeList(int z){
        while((int)prime_list.size() < z){
            if((int)isprime.size() <= prime_list_max + 1) CalcIsPrime(prime_list_max + 1);
            for(int p=prime_list_max+1; p<(int)isprime.size(); p++){
                if(isprime[p]) prime_list.push_back(p*2+1);
            }
            prime_list_max = isprime.size() - 1;
        }
    }

    void CalcPrimeListUntil(int z){
        if(prime_list_max < z){
            CalcIsPrime(z);
            for(int p=prime_list_max+1; p<(int)isprime.size(); p++){
                if(isprime[p]) prime_list.push_back(p*2+1);
            }
            prime_list_max = isprime.size() - 1;
        }
    }

}


bool IsprimeExplicit(int n){
    using namespace prime_sieve_explicit_internal;
    if(n == 2) return true;
    if(n % 2 == 0) return false;
    CalcIsPrime(n);
    return isprime[(n-1)/2];
}

int NthPrimeExplicit(int n){
    using namespace prime_sieve_explicit_internal;
    CalcPrimeList(n);
    return prime_list[n];
}

int PrimeCountingExplicit(int n){
    using namespace prime_sieve_explicit_internal;
    if(n < 2) return 0;
    CalcPrimeListUntil(n);
    auto res = std::upper_bound(prime_list.begin(), prime_list.end(), n) - prime_list.begin();
    return (int)res;
}

// [l, r)
std::vector<bool> SegmentedSieveExplicit(long long l, long long r){
    assert(0 <= l); assert(l <= r);
    long long d = r - l;
    if(d == 0) return {};
    std::vector<bool> res(d, true);
    for(long long p=2; p*p<=r; p++) if(IsprimeExplicit(p)){
        long long il = (l+p-1)/p, ir = (r+p-1)/p;
        if(il <= p) il = p;
        for(long long i=il; i<ir; i++) res[i*p-l] = false;
    }
    if(l < 2) for(long long p=l; p<2 && p<r; p++) res[l-p] = false;
    return res;
}


} // namespace nachia
#line 4 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"

#line 8 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"

namespace nachia{

template<class Elem>
void DivisorZeta(std::vector<Elem>& a){
    using namespace prime_sieve_explicit_internal;
    int n = a.size() - 1;
    for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=1; i*d<=n; i++) a[i*d] += a[i];
}

template<class Elem>
void DivisorInvZeta(std::vector<Elem>& a){
    using namespace prime_sieve_explicit_internal;
    int n = a.size() - 1;
    for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=n/d; i>=1; i--) a[i] += a[i*d];
}

template<class Elem>
void DivisorMobius(std::vector<Elem>& a){
    using namespace prime_sieve_explicit_internal;
    int n = a.size() - 1;
    for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=n/d; i>=1; i--) a[i*d] -= a[i];
}

template<class Elem>
void DivisorInvMobius(std::vector<Elem>& a){
    using namespace prime_sieve_explicit_internal;
    int n = a.size() - 1;
    for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=1; i*d<=n; i++) a[i] -= a[i*d];
}

template<class Elem>
std::vector<Elem> GcdConvolution(std::vector<Elem> a, std::vector<Elem> b){
    assert(a.size() == b.size());
    assert(1 <= a.size());
    DivisorInvZeta(a);
    DivisorInvZeta(b);
    for(int i=1; i<(int)a.size(); i++) a[i] *= b[i];
    DivisorInvMobius(a);
    return a;
}

template<class Elem>
std::vector<Elem> LcmConvolution(std::vector<Elem> a, std::vector<Elem> b){
    assert(a.size() == b.size());
    assert(1 <= a.size());
    DivisorZeta(a);
    DivisorZeta(b);
    for(int i=1; i<(int)a.size(); i++) a[i] *= b[i];
    DivisorMobius(a);
    return a;
}

}
#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\floor-sum.hpp"

#line 4 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\floor-sum.hpp"
#include <utility>

namespace nachia{


// a : any value
// mod != 0
std::pair<long long, unsigned long long> SafeDiv(long long a, unsigned long long mod){
    using u64 = unsigned long long;
    if(a >= 0) return std::make_pair(0, (u64)a);
    if(mod >= (u64)1 << 62) return std::make_pair(-1, (u64)a + mod);
    long long q = a / mod;
    long long m = a % (long long)mod;
    if(m){ q--; m += mod; }
    return std::make_pair(q, m);
}

unsigned long long nC2Uint64(unsigned long long n){
    return (n%2) ? ((n-1)/2*n) : (n/2*(n-1));
}

// n : any
// 1 <= m
// a : any
// b : any
// n * a%m + b%m < 2**64
unsigned long long FloorSumU64Unsigned(
    unsigned long long n,
    unsigned long long m,
    unsigned long long a,
    unsigned long long b
){
    using u64 = unsigned long long;
    assert(1 <= m);
    u64 ans = 0;
    while(n){
        if(a >= m){
            ans += a / m * nC2Uint64(n);
            a %= m;
        }
        if(b >= m){
            ans += b / m * n;
            b %= m;
        }
        u64 y_max = a * n + b;
        if (y_max < m) return ans;
        n = y_max / m;
        b = y_max % m;
        y_max = a; a = m; m = y_max;
    }
    return ans;
}

// n : any
// 1 <= m
// a : any
// b : any
// (n+1) * m < 2**64
unsigned long long FloorSumU64Signed(
    unsigned long long n,
    unsigned long long m,
    long long a,
    long long b
){

    using u64 = unsigned long long;
    auto ua = SafeDiv(a, m);
    auto ub = SafeDiv(b, m);
    u64 ans = FloorSumU64Unsigned(n, m, ua.second, ub.second);
    ans += ua.first / m * nC2Uint64(n);
    ans += ub.first / m * n;
    return ans;
}

} // namespace nachia
#line 4 "..\\Main.cpp"
#include <string>
#line 7 "..\\Main.cpp"
#include <atcoder/modint>
using namespace std;
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
#define rep(i,n) for(int i=0; i<(int)(n); i++)
const i64 INF = 1001001001001001001;

using Modint = atcoder::static_modint<998244353>;

i64 countFracs(i64 N){
    vector<i64> cnt(N+1);
    for(i64 c=1; c<=N; c++) cnt[c] += c;
    nachia::DivisorMobius(cnt);
    i64 sumcnt = 0;
    for(int i=1; i<=N; i++) sumcnt += cnt[i];
    return sumcnt;
}

pair<i64,i64> testcase(){
    i64 N, K; cin >> N >> K;
    i64 cnt = countFracs(N);
    if(cnt * 2 - 1 < K) return {-1,-1};
    if(cnt == K) return {1,1};
    bool sw = false;
    if(cnt < K){ K = cnt*2 - K; sw = true; }
    i64 l = 0, r = 1ll << 40;

    vector<i64> quotients = {0};
    for(i64 k=1; k*k<N; k++) quotients.push_back(k);
    int qp1 = quotients.size();
    for(i64 k=1; k*k<=N; k++) quotients.push_back(N/k);
    reverse(quotients.begin() + qp1, quotients.end());
    int numQ = quotients.size()-1;

    vector<i64> mobius(N+1); mobius[1] = 1;
    nachia::DivisorMobius(mobius);
    vector<i64> mertens = mobius;
    rep(i,N) mertens[i+1] += mertens[i];

    while(l+1 < r){
        i64 m = (l+r)/2;
        i64 sumcnt = 0;
        /*
        */
        rep(q,numQ){
            i64 times = mertens[quotients[q+1]] - mertens[quotients[q]];
            cnt = nachia::FloorSumU64Signed(quotients[numQ-q]+1, 1ll << 40, m, 0);
            sumcnt += times * cnt;
        }
        //vector<i64> cnt(N+1);
        //for(i64 a=1; a<=N; a++) cnt[a] += (m * a) >> 40; // floor(m*a/b)
        //nachia::DivisorMobius(cnt);
        //for(int i=1; i<=N; i++) sumcnt += cnt[i];
        if(sumcnt < K) l = m; else r = m;
    }
    // l < x <= r
    if(l == (1ll << 40)) return {-1,-1};
    i64 a = N+1, b = 1;
    for(int i=1; i<=N; i++){
        i64 q = ((l * i) >> 40) + 1;
        if(q > N) continue;
        if(q * b < a * i){ a = q; b = i; }
    }
    if(sw) swap(a, b);
    return {a,b};
}

int main(){
    int T; cin >> T;
    rep(t,T){
        auto ans = testcase();
        if(ans.first < 0) cout << "-1\n"; else cout << ans.first << '/' << ans.second << '\n';
    }
    return 0;
}



struct ios_do_not_sync{
    ios_do_not_sync(){
        ios::sync_with_stdio(false);
        cin.tie(nullptr);
    }
} ios_do_not_sync_instance;
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