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:heavy_check_mark: Verify/YUKI_1781.test.cpp

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Code

#define PROBLEM "https://yukicoder.me/problems/no/1781"

#include "Template/template.hpp"

#include "Utility/fastio.hpp"


#include "Math/multiplicative.hpp"

#include "Math/modint.hpp"

#include "Math/primesum.hpp"

ll F(ll x){return x;}
PrimeSum<ll,F> buf;

using Fp=fp<998244353>;
Fp memo[50];
Fp pe(int p,int e){return memo[e];}
Fp psum(ll x){return memo[1]*buf[x];}

int main(){
    ll n,m;
    read(n,m);

    rep(e,0,45)memo[e]=Fp(e+1).pow(n);
    buf=PrimeSum<ll,F>(m);

    auto ret=MultiplicativeSum<Fp,pe,psum>(m);
    print(ret.v);
    return 0;
}
#line 1 "Verify/YUKI_1781.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/1781"

#line 1 "Template/template.hpp"
#include <bits/stdc++.h>

using namespace std;

#define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b-1); i >= (int)(a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())

using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
const int inf = 0x3fffffff;
const ll INF = 0x1fffffffffffffff;

template <typename T> inline bool chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return 1;
    }
    return 0;
}
template <typename T> inline bool chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return 1;
    }
    return 0;
}
template <typename T, typename U> T ceil(T x, U y) {
    assert(y != 0);
    if (y < 0)
        x = -x, y = -y;
    return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U> T floor(T x, U y) {
    assert(y != 0);
    if (y < 0)
        x = -x, y = -y;
    return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T> int popcnt(T x) {
    return __builtin_popcountll(x);
}
template <typename T> int topbit(T x) {
    return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <typename T> int lowbit(T x) {
    return (x == 0 ? -1 : __builtin_ctzll(x));
}

#ifdef LOCAL
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define show(...) true
#endif
template <typename T> void _show(int i, T name) {
    cerr << '\n';
}
template <typename T1, typename T2, typename... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
    for (; a[i] != ',' && a[i] != '\0'; i++)
        cerr << a[i];
    cerr << ":" << b << " ";
    _show(i + 1, a, c...);
}
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
    os << "P(" << p.first << ", " << p.second << ")";
    return os;
}
template <typename T, template <class> class C>
ostream &operator<<(ostream &os, const C<T> &v) {
    os << "[";
    for (auto d : v)
        os << d << ", ";
    os << "]";
    return os;
}
#line 2 "Utility/fastio.hpp"
#include <unistd.h>

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() {
    memmove(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 <typename T> void rd_real(T &x) {
    string s;
    rd(s);
    x = stod(s);
}

template <typename T> 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<T>::value || is_same_v<T, i128>) {
        if (c == '-') {
            minus = 1, c = ibuf[pil++];
        }
    }
    x = 0;
    while ('0' <= c) {
        x = x * 10 + (c & 15), c = ibuf[pil++];
    }
    if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
        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(uint &x) {
    rd_integer(x);
}
void rd(ull &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);
}

template <class T, class U> void rd(pair<T, U> &p) {
    return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T> void rd_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
        auto &x = std::get<N>(t);
        rd(x);
        rd_tuple<N + 1>(t);
    }
}
template <class... T> void rd(tuple<T...> &tpl) {
    rd_tuple(tpl);
}

template <size_t N = 0, typename T> void rd(array<T, N> &x) {
    for (auto &d : x)
        rd(d);
}
template <class T> void rd(vector<T> &x) {
    for (auto &d : x)
        rd(d);
}

void read() {}
template <class H, class... T> 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 <typename T> 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 <typename T> 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(uint x) {
    wt_integer(x);
}
void wt(ull 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);
}

template <class T, class U> void wt(const pair<T, U> val) {
    wt(val.first);
    wt(' ');
    wt(val.second);
}
template <size_t N = 0, typename T> void wt_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
        if constexpr (N > 0) {
            wt(' ');
        }
        const auto x = std::get<N>(t);
        wt(x);
        wt_tuple<N + 1>(t);
    }
}
template <class... T> void wt(tuple<T...> tpl) {
    wt_tuple(tpl);
}
template <class T, size_t S> void wt(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
        if (i)
            wt(' ');
        wt(val[i]);
    }
}
template <class T> void wt(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
        if (i)
            wt(' ');
        wt(val[i]);
    }
}

void print() {
    wt('\n');
}
template <class Head, class... Tail> void print(Head &&head, Tail &&...tail) {
    wt(head);
    if (sizeof...(Tail))
        wt(' ');
    print(forward<Tail>(tail)...);
}
void __attribute__((destructor)) _d() {
    flush();
}
} // namespace fastio


using fastio::flush;
using fastio::print;
using fastio::read;

inline void first(bool i = true) {
    print(i ? "first" : "second");
}
inline void Alice(bool i = true) {
    print(i ? "Alice" : "Bob");
}
inline void yes(bool i = true) {
    print(i ? "yes" : "no");
}
inline void Yes(bool i = true) {
    print(i ? "Yes" : "No");
}
inline void No() {
    print("No");
}
inline void YES(bool i = true) {
    print(i ? "YES" : "NO");
}
inline void NO() {
    print("NO");
}
inline void Yay(bool i = true) {
    print(i ? "Yay!" : ":(");
}
inline void Possible(bool i = true) {
    print(i ? "Possible" : "Impossible");
}
inline void POSSIBLE(bool i = true) {
    print(i ? "POSSIBLE" : "IMPOSSIBLE");
}

/**
 * @brief Fast IO
 */
#line 5 "Verify/YUKI_1781.test.cpp"

#line 2 "Math/sieve.hpp"

template<int L=50101010>vector<int> sieve(int N){
    bitset<L> isp;
    int n,sq=ceil(sqrt(N));
    for(int z=1;z<=5;z+=4){
        for(int y=z;y<=sq;y+=6){
            for(int x=1;x<=sq and (n=4*x*x+y*y)<=N;++x){
                isp[n].flip();
            }
            for(int x=y+1;x<=sq and (n=3*x*x-y*y)<=N;x+=2){
                isp[n].flip();
            }
        }
    }
    for(int z=2;z<=4;z+=2){
        for(int y=z;y<=sq;y+=6){
            for (int x=1;x<=sq and (n=3*x*x+y*y)<=N;x+=2){
                isp[n].flip();
            }
            for(int x=y+1;x<=sq and (n=3*x*x-y*y)<=N;x+=2){
                isp[n].flip();
            }
        }
    }
    for(int y=3;y<=sq;y+=6){
        for(int z=1;z<=2;++z){
            for(int x=z;x<=sq and (n=4*x*x+y*y)<=N;x+=3){
                isp[n].flip();
            }
        }
    }
    for(int n=5;n<=sq;++n)if(isp[n]){
        for(int k=n*n;k<=N;k+=n*n){
            isp[k]=false;
        }
    }
    isp[2]=isp[3]=true;

    vector<int> ret;
    for(int i=2;i<=N;i++)if(isp[i]){
        ret.push_back(i);
    }
    return ret;
}

/**
 * @brief Prime Sieve
 */
#line 3 "Math/multiplicative.hpp"

template<typename T,T (*pe)(int,int),T (*psum)(ll)>T MultiplicativeSum(ll N){
    ll SQ=sqrtl(N);
    auto ps=sieve(SQ);
    
    T ret=psum(N)+1;
    auto dfs=[&](auto& dfs,ll x,int i,int e,T cur,T pre)->void{
        T nxt=pre*pe(ps[i],e+1);
        ret+=cur*(psum(double(N)/x)-psum(ps[i]));
        ret+=nxt;
        ll L=sqrtl(double(N)/x);
        if(ps[i]<=L)dfs(dfs,x*ps[i],i,e+1,nxt,pre);
        rep(j,i+1,ps.size()){
            if(ps[j]>L)break;
            dfs(dfs,x*ps[j],j,1,cur*pe(ps[j],1),cur);
        }
    };
    rep(i,0,ps.size())dfs(dfs,ps[i],i,1,pe(ps[i],1),1);
    return ret;
}

template<typename T,T (*pe)(int,int),ll (*pcnt)(ll),T (*psum)(ll)>T AdditiveSum(ll N){
    ll SQ=sqrtl(N);
    auto ps=sieve(SQ);
    
    T ret=psum(N);
    auto dfs=[&](auto& dfs,ll x,int i,int e,T cur,T pre)->void{
        T nxt=pre+pe(ps[i],e+1);
        ret+=cur*(pcnt(double(N)/x)-pcnt(ps[i]))+(psum(double(N)/x)-psum(ps[i]));
        ret+=nxt;
        ll L=sqrtl(double(N)/x);
        if(ps[i]<=L)dfs(dfs,x*ps[i],i,e+1,nxt,pre);
        rep(j,i+1,ps.size()){
            if(ps[j]>L)break;
            dfs(dfs,x*ps[j],j,1,cur+pe(ps[j],1),cur);
        }
    };
    rep(i,0,ps.size())dfs(dfs,ps[i],i,1,pe(ps[i],1),0);
    return ret;
}

/**
 * @brief Multiplicative Sum
 * @docs docs/multiplicative.md
 */
#line 2 "Math/modint.hpp"

template <unsigned mod = 1000000007> struct fp {
    unsigned v;
    static constexpr int get_mod() {
        return mod;
    }
    constexpr unsigned inv() const {
        assert(v != 0);
        int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
        while (y > 0) {
            t = x / y;
            x -= t * y, p -= t * q;
            tmp = x, x = y, y = tmp;
            tmp = p, p = q, q = tmp;
        }
        if (p < 0)
            p += mod;
        return p;
    }
    constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
    fp operator-() const {
        return fp() - *this;
    }
    fp pow(ull t) {
        fp res = 1, b = *this;
        while (t) {
            if (t & 1)
                res *= b;
            b *= b;
            t >>= 1;
        }
        return res;
    }
    fp &operator+=(const fp &x) {
        if ((v += x.v) >= mod)
            v -= mod;
        return *this;
    }
    fp &operator-=(const fp &x) {
        if ((v += mod - x.v) >= mod)
            v -= mod;
        return *this;
    }
    fp &operator*=(const fp &x) {
        v = ull(v) * x.v % mod;
        return *this;
    }
    fp &operator/=(const fp &x) {
        v = ull(v) * x.inv() % mod;
        return *this;
    }
    fp operator+(const fp &x) const {
        return fp(*this) += x;
    }
    fp operator-(const fp &x) const {
        return fp(*this) -= x;
    }
    fp operator*(const fp &x) const {
        return fp(*this) *= x;
    }
    fp operator/(const fp &x) const {
        return fp(*this) /= x;
    }
    bool operator==(const fp &x) const {
        return v == x.v;
    }
    bool operator!=(const fp &x) const {
        return v != x.v;
    }
    friend istream &operator>>(istream &is, fp &x) {
        return is >> x.v;
    }
    friend ostream &operator<<(ostream &os, const fp &x) {
        return os << x.v;
    }
};

template <unsigned mod> void rd(fp<mod> &x) {
    fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
    fastio::wt(x.v);
}

template <typename T> T Inv(ll n) {
    static const int md = T::get_mod();
    static vector<T> buf({0, 1});
    assert(n > 0);
    n %= md;
    while (SZ(buf) <= n) {
        int k = SZ(buf), q = (md + k - 1) / k;
        buf.push_back(buf[k * q - md] * q);
    }
    return buf[n];
}

template <typename T> T Fact(ll n, bool inv = 0) {
    static const int md = T::get_mod();
    static vector<T> buf({1, 1}), ibuf({1, 1});
    assert(n >= 0 and n < md);
    while (SZ(buf) <= n) {
        buf.push_back(buf.back() * SZ(buf));
        ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
    }
    return inv ? ibuf[n] : buf[n];
}

template <typename T> T nPr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nHr(int n, int r, bool inv = 0) {
    return nCr<T>(n + r - 1, r, inv);
}

/**
 * @brief Modint
 */
#line 3 "Math/primesum.hpp"

template<typename T,T (*F)(ll)>struct PrimeSum{
    ll N,SQ;
    vector<T> lo,hi;
    PrimeSum(ll n=0):N(n),SQ(sqrtl(N)),lo(SQ+1),hi(SQ+1){
        rep(i,1,SQ+1){
            lo[i]=F(i)-1;
            hi[i]=F(N/i)-1;
        }
        auto ps=sieve(SQ);
        for(auto& p:ps){
            ll q=ll(p)*p;
            if(q>N)break;
            T sub=lo[p-1],fp=lo[p]-lo[p-1];
            ll L=min(SQ,N/q),M=SQ/p;
            rep(i,1,M+1)hi[i]-=fp*(hi[i*p]-sub);
            rep(i,M+1,L+1)hi[i]-=fp*(lo[double(N)/(i*p)]-sub);
            for(int i=SQ;i>=q;i--)lo[i]-=fp*(lo[double(i)/p]-sub);
        }
    }
    T operator[](ll x) {
        return (x<=SQ?lo[x]:hi[N/x]);
    }
};

/**
 * @brief Prime Sum
 * @docs docs/primesum.md
 */
#line 9 "Verify/YUKI_1781.test.cpp"
ll F(ll x){return x;}
PrimeSum<ll,F> buf;

using Fp=fp<998244353>;
Fp memo[50];
Fp pe(int p,int e){return memo[e];}
Fp psum(ll x){return memo[1]*buf[x];}

int main(){
    ll n,m;
    read(n,m);

    rep(e,0,45)memo[e]=Fp(e+1).pow(n);
    buf=PrimeSum<ll,F>(m);

    auto ret=MultiplicativeSum<Fp,pe,psum>(m);
    print(ret.v);
    return 0;
}
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