CP-Algorithms Library

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:heavy_check_mark: Enumerate Quotients (verify/number_theory/quotients.test.cpp)

Depends on

Code

// @brief Enumerate Quotients
#define PROBLEM "https://judge.yosupo.jp/problem/enumerate_quotients"
#pragma GCC optimize("O3,unroll-loops")
#include <iostream>
#include "blazingio/blazingio.min.hpp"
#include "cp-algo/number_theory/dirichlet.hpp"
#include <bits/stdc++.h>

using namespace std;
using namespace cp_algo::math;

void solve() {
    int64_t n;
    cin >> n;
    auto res = floors(n) | views::drop(1);
    cout << size(res) << "\n";
    for(auto it: res) {
        cout << it << " ";
    }
    cout << "\n";
}

signed main() {
    //freopen("input.txt", "r", stdin);
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t = 1;
    //cin >> t;
    while(t--) {
        solve();
    }
}

#line 1 "verify/number_theory/quotients.test.cpp"
// @brief Enumerate Quotients
#define PROBLEM "https://judge.yosupo.jp/problem/enumerate_quotients"
#pragma GCC optimize("O3,unroll-loops")
#include <iostream>
#line 1 "blazingio/blazingio.min.hpp"
// NOLINTBEGIN
// clang-format off
// DO NOT REMOVE THIS MESSAGE. The mess that follows is a minified build of
// https://github.com/purplesyringa/blazingio. Refer to the repository for
// a human-readable version and documentation.
// Options: cbfoiedrhWLMXaIaAn
#define M$(x,...)_mm256_##x##_epi8(__VA_ARGS__)
#define $u(...)__VA_ARGS__
#if __APPLE__
#define $m(A,B)A
#else
#define $m(A,B)B
#endif
#if _WIN32
#define $w(A,B)A
#else
#define $w(A,B)B
#endif
#if __i386__|_M_IX86
#define $H(A,B)A
#else
#define $H(A,B)B
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#if __aarch64__
#define $a(A,B)A
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#define $a(A,B)B
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#define $P(x)void F(x K){
#define $T template<$c T
#define $c class
#define $C constexpr
#define $R return
#define $O operator
#define u$ uint64_t
#define $r $R*this;
#include<array>
#include<bitset>
#include<complex>
#include<cstring>
#include $a(<arm_neon.h>,<immintrin.h>)
#include<stdint.h>
#include $w(<windows.h>,<sys/mman.h>)
#include<sys/stat.h>
#include $w(<io.h>,<unistd.h>)
#include $w(<ios>,<sys/resource.h>)
#if _MSC_VER
#define __builtin_add_overflow(a,b,c)_addcarry_u64(0,a,b,c)
#define $s
#else
$H(,u$ _umul128(u$ a,u$ b,u$*D){auto x=(__uint128_t)a*b;*D=u$(x>>64);$R(u$)x;})
#define $s $a(,__attribute__((target("avx2"))))
#endif
#define $z $a(16,32)
#define $t $a(uint8x16_t,__m256i)
#define $I $w(__forceinline,__attribute__((always_inline)))
#define $F M(),
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V{char*D;B*S;int J;V(){$w(E$(D=(char*)VirtualAlloc(0,536870912,8192,4))E$(VirtualAlloc(D,4096,4096,260))AddVectoredExceptionHandler(1,$x);,size_t C=536870912;$m(,rlimit E;getrlimit(RLIMIT_AS,&E);if(~E.rlim_cur)C=25165824;)D=(char*)mmap(0,C,3,$m(4162,16418),-1,0);E$(D!=(void*)-1))S=(B*)D;for(int i=0;i<100;i++)A[i]=short((48+i/10)|((48+i%10)<<8));for(int i=1;i<64;i++)L[i]=L[i-1]+(0x8922489224892249>>i&1);}~V(){flush($w(!J,));}void flush($w(int F=0,)){$w(J=1;auto E=GetStdHandle(-11);auto C=F?ReOpenFile(E,1073741824,7,2684354560):(void*)-1;DWORD A;E$(C==(void*)-1?WriteFile(E,D,DWORD((char*)S-D),&A,0):(WriteFile(C,D,DWORD(((char*)S-D+4095)&-4096),&A,0)&&~_chsize(1,int((char*)S-D)))),auto G=D;ssize_t A;while((A=write(1,G,(char*)S-G))>0)G+=A;E$(~A))S=(B*)D;}$P(char)*S++=K;}$P(uint8_t)*S++=K;}$P(int8_t)*S++=K;}$P(bool)*S++=48+K;}$T>decltype((void)~T{1})F(T K){using D=make_unsigned_t<T>;D C=K;if(K<0)F('-'),C=1+~C;static $C auto N=[](){array<D,5*sizeof(T)/2>N{};D n=1;for(size_t i=1;i<N.size();i++)n*=10,N[i]=n;$R N;}();$w(ULONG M;,)int G=L[$w(($H(_BitScanReverse(&M,ULONG((int64_t)C>>32))?M+=32:_BitScanReverse(&M,(ULONG)C|1),_BitScanReverse64(&M,C|1)),M),63^__builtin_clzll(C|1))];G-=C<N[G-1];short H[20];if $C(sizeof(T)==2){auto n=33555U*C-C/2;u$ H=A[n>>25];n=(n&33554431)*25;H|=A[n>>23]<<16;H|=u$(48+((n&8388607)*5>>22))<<32;H>>=40-G*8;memcpy(S,&H,8);}else if $C(sizeof(T)==4){auto n=1441151881ULL*C;$H(n>>=25;n++;for(int i=0;i<5;i++){H[i]=A[n>>32];n=(n&~0U)*100;},int K=57;auto J=~0ULL>>7;for(int i=0;i<5;i++){H[i]=A[n>>K];n=(n&J)*25;K-=2;J/=4;})memcpy(S,(B*)H+10-G,16);}else{$H($u(if(C<(1ULL<<32)){$R F((uint32_t)C);}auto J=(u$)1e10;auto x=C/J,y=C%J;int K=100000,b[]{int(x/K),int(x%K),int(y/K),int(y%K)};B H[40];for(int i=0;i<4;i++){int n=int((429497ULL*b[i]>>7)+1);B*p=H+i*5;*p=48+char(n>>25);n=(n&~0U>>7)*25;memcpy(p+1,A+(n>>23),2);memcpy(p+3,A+((n&~0U>>9)*25>>21),2);}),$u(u$ D,E=_umul128(18,C,&D),F;_umul128(0x725dd1d243aba0e8,C,&F);D+=__builtin_add_overflow(E,F+1,&E);for(int i=0;i<10;i++)H[i]=A[D],E=_umul128(100,E,&D);))memcpy(S,(B*)H+20-G,20);}S+=G;}$T>decltype((void)T{1.})F(T K){if(K<0)F('-'),K=-K;auto G=[&](){auto x=u$(K*1e12);$H($u(x-=x>999999999999;uint32_t n[]{uint32_t(x/1000000*429497>>7)+1,uint32_t(x%1000000*429497>>7)+1};int K=25,J=~0U>>7;for(int i=0;i<3;i++){for(int j=0;j<2;j++)memcpy(S+i*2+j*6,A+(n[j]>>K),2),n[j]=(n[j]&J)*25;K-=2;J/=4;}S+=12;),$u(u$ D,E=_umul128(472236648287,x,&D)>>8;E|=D<<56;D>>=8;E++;for(int i=0;i<6;i++)memcpy(S,A+D,2),S+=2,E=_umul128(100,E,&D);))};if(K==0)$R F('0');if(K>=1e16){K*=(T)1e-16;int B=16;while(K>=1)K*=(T).1,B++;F("0.");G();F('e');F(B);}else if(K>=1){auto B=(u$)K;F(B);if((K-=(T)B)>0)F('.'),G();}else F("0."),G();}$P(const char*)$w(size_t A=strlen(K);memcpy((char*)S,K,A);S+=A;,S=(B*)stpcpy((char*)S,K);)}$P(const uint8_t*)F((char*)K);}$P(const int8_t*)F((char*)K);}$P(string_view)memcpy(S,K.data(),K.size());S+=K.size();}$T>$P(complex<T>)*this<<'('<<K.real()<<','<<K.imag()<<')';}template<size_t N>$s $P(const bitset<N>&)auto i=N;while(i%$z)*S++=48+K[--i];B*p=S;while(i){i-=$z;$a(short,int)x;memcpy(&x,(char*)&K+i/8,$z/8);$a(auto A=(uint8x8_t)vdup_n_u16(x);vst1q_u8((uint8_t*)p,48-vtstq_u8(vcombine_u8(vuzp2_u8(A,A),vuzp1_u8(A,A)),(uint8x16_t)vdupq_n_u64(~2ULL/254)));,auto b=_mm256_set1_epi64x(~2ULL/254);_mm256_storeu_si256(($t*)p,M$(sub,M$(set1,48),M$(cmpeq,_mm256_and_si256(M$(shuffle,_mm256_set1_epi32(x),_mm256_set_epi64x(0,C,C*2,C*3)),b),b)));)p+=$z;}S=p;}$T>V&$O<<(const T&K){F(K);$r}V&$O<<(V&(*A)(V&)){$R A(*this);}};struct W{$T>W&$O<<(const T&K){$r}W&$O<<(W&(*A)(W&)){$R A(*this);}};}namespace std{$f::U i$;$f::V o$;$f::W e$;$f::U&getline($f::U&B,string&K){$f::D A{K};$R B>>A;}$f::V&flush($f::V&B){if(!i$.A.S)B.flush();$R B;}$f::V&endl($f::V&B){$R B<<'\n'<<flush;}$f::W&endl($f::W&B){$R B;}$f::W&flush($f::W&B){$R B;}}$w(LONG WINAPI $x(_EXCEPTION_POINTERS*A){auto C=A->ExceptionRecord;auto B=C->ExceptionInformation[1];if(C->ExceptionCode==2147483649&&B-(ULONG_PTR)std::o$.D<0x40000000){E$(VirtualAlloc((char*)B,16777216,4096,4)&&VirtualAlloc((char*)(B+16777216),4096,4096,260))$R-1;}$R 0;},)
#define freopen(...)if(freopen(__VA_ARGS__)==stdin)std::i$=$f::U{}
#define cin i$
#define cout o$
#ifdef ONLINE_JUDGE
#define cerr e$
#define clog e$
#endif
// End of blazingio
// NOLINTEND
// clang-format on
#line 1 "cp-algo/number_theory/dirichlet.hpp"


#include <algorithm>
#include <cstdint>
#include <ranges>
#include <vector>
#include <cmath>
namespace cp_algo::math {    
    auto floor_stats(int64_t n) {
        auto rt_n = int(sqrtl(n));
        return std::pair{rt_n, 2 * rt_n - (n / rt_n == rt_n)};
    }

    auto floor_generator(int64_t n) {
        auto [rt_n, num_floors] = floor_stats(n);
        return [n, rt_n = rt_n, num_floors = num_floors](int k) {
            return k <= rt_n ? int64_t(k) : n / int64_t(num_floors - k + 1);
        };
    }

    auto floors(int64_t n) {
        auto [_, m] = floor_stats(n);
        return std::views::iota(0, m+1) | std::views::transform(floor_generator(n));
    }

    struct interval {
        int lo, hi;
        auto operator <=>(const interval&) const = default;
    };

    // callback(k) when:
    //     (F * G)[k] = H[k] + (F[k] - F[k-1]) * G[1] + (G[k] - G[k-1]) * F[1]
    // Return the value to be saved in H[k]
    enum exec_mode { standard, reverse };
    template<exec_mode mode = standard>
    void Dirichlet_helper(int64_t n, auto &H, auto const& F, auto const& G, auto &&callback) {
        auto [rt_n, num_floors] = floor_stats(n);

        auto to_ord = [&](int64_t k) {
            return k <= rt_n ? int(k) : num_floors - int(n / k) + 1;
        };

        auto call = [&](interval x, interval y, interval z) {
            auto Fx = F[x.hi] - F[x.lo - 1];
            auto Fy = F[y.hi] - F[y.lo - 1];
            decltype(Fx) Gx, Gy;
            if constexpr (mode == standard) {
                Gy = G[y.hi] - G[y.lo - 1];
                Gx = G[x.hi] - G[x.lo - 1];
            } else {
                Gy = G[y.lo - 1] - G[y.hi];
                Gx = G[x.lo - 1] - G[x.hi];
            }
            auto t = Fx * Gy;
            if(x != y) [[likely]] {
                t += Fy * Gx;
            }
            H[z.lo] += t;
            if (z.hi < num_floors) [[likely]] {
                H[z.hi + 1] -= t;
            }
        };
        for (int k = 2; k <= num_floors; ++k) {
            if(k > rt_n) {
                int z = num_floors - k + 1;
                for (int x = 2; ; x++) {
                    int y_lo_ord = std::max(x, z) + 1;
                    int y_hi_ord = to_ord(n / (x * z));
                    if (y_hi_ord < y_lo_ord) break;
                    call({x, x}, {y_lo_ord, y_hi_ord}, {k, k});
                }
            }

            H[k] = callback(k);

            if(k <= rt_n) {
                int x = k;
                for (int y = 2; y < k; ++y) {
                    int z_lo_ord = to_ord(1LL * x * y);
                    int z_hi_ord = to_ord(n / x);
                    if (z_hi_ord < z_lo_ord) break;
                    call({x, x}, {y, y}, {z_lo_ord, z_hi_ord});
                }
                int z_lo_ord = to_ord(1LL * x * x);
                call({x, x}, {x, x}, {z_lo_ord, num_floors});
            }
        }
    }

    auto Dirichlet_mul(auto const& F, auto const& G, int64_t n) {
        auto m = size(F);
        std::decay_t<decltype(F)> H(m);
        H[1] = F[1] * G[1];
        Dirichlet_helper(n, H, F, G, [&](auto k) {
            return H[k] + (F[k] - F[k-1]) * G[1] + (G[k] - G[k-1]) * F[1];
        });
        partial_sum(begin(H), end(H), begin(H));
        return H;
    }

    void Dirichlet_div_inplace(auto &H, auto const& G, int64_t n) {
        auto Gi = G[1].inv();
        H[0] -= H[0];
        adjacent_difference(begin(H), end(H), begin(H));
        H[1] *= Gi;
        Dirichlet_helper<reverse>(n, H, H, G, [&](auto k) {
            return (Gi * (H[k] - (G[k] - G[k-1]) * H[1])) + H[k-1];
        });
    }

    auto Dirichlet_div(auto const& H, auto const& G, int64_t n) {
        auto m = std::size(G);
        auto F = H | std::views::take(m) | std::ranges::to<std::vector>();
        Dirichlet_div_inplace(F, G, n);
        return F;
    }
}

#line 7 "verify/number_theory/quotients.test.cpp"
#include <bits/stdc++.h>

using namespace std;
using namespace cp_algo::math;

void solve() {
    int64_t n;
    cin >> n;
    auto res = floors(n) | views::drop(1);
    cout << size(res) << "\n";
    for(auto it: res) {
        cout << it << " ";
    }
    cout << "\n";
}

signed main() {
    //freopen("input.txt", "r", stdin);
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t = 1;
    //cin >> t;
    while(t--) {
        solve();
    }
}

Test cases

Env Name Status Elapsed Memory
g++ example_00 :heavy_check_mark: AC 4 ms 6 MB
g++ max_00 :heavy_check_mark: AC 40 ms 20 MB
g++ min_00 :heavy_check_mark: AC 4 ms 6 MB
g++ min_01 :heavy_check_mark: AC 4 ms 6 MB
g++ min_02 :heavy_check_mark: AC 4 ms 6 MB
g++ min_03 :heavy_check_mark: AC 3 ms 6 MB
g++ min_04 :heavy_check_mark: AC 3 ms 6 MB
g++ min_05 :heavy_check_mark: AC 3 ms 6 MB
g++ min_06 :heavy_check_mark: AC 3 ms 6 MB
g++ min_07 :heavy_check_mark: AC 3 ms 6 MB
g++ min_08 :heavy_check_mark: AC 3 ms 6 MB
g++ polynom_p1p1m1_00 :heavy_check_mark: AC 41 ms 20 MB
g++ polynom_p1p1m1_01 :heavy_check_mark: AC 21 ms 12 MB
g++ polynom_p1p1z_00 :heavy_check_mark: AC 40 ms 20 MB
g++ polynom_p1p1z_01 :heavy_check_mark: AC 21 ms 12 MB
g++ polynom_p1zm1_00 :heavy_check_mark: AC 39 ms 20 MB
g++ polynom_p1zm1_01 :heavy_check_mark: AC 21 ms 12 MB
g++ random_00 :heavy_check_mark: AC 21 ms 12 MB
g++ random_01 :heavy_check_mark: AC 28 ms 14 MB
g++ random_small_00 :heavy_check_mark: AC 4 ms 6 MB
g++ random_small_01 :heavy_check_mark: AC 4 ms 6 MB
g++ random_small_02 :heavy_check_mark: AC 3 ms 6 MB
g++ random_small_03 :heavy_check_mark: AC 3 ms 6 MB
g++ random_small_04 :heavy_check_mark: AC 3 ms 6 MB
g++ square_00 :heavy_check_mark: AC 17 ms 10 MB
g++ square_01 :heavy_check_mark: AC 20 ms 12 MB
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