noya2_Library

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:heavy_check_mark: test/math/SumofMultiplicativeFunction.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_multiplicative_function"

#include"../../template/template.hpp"
#include"../../utility/modint.hpp"
using mint = static_modint<469762049>;
using ar = array<mint,2>;
ar &operator+=(ar &a, ar b){
    a[0] += b[0];
    a[1] += b[1];
    return a;
}
ar &operator-=(ar &a, ar b){
    a[0] -= b[0];
    a[1] -= b[1];
    return a;
}
ar operator-(ar a, ar b){
    return a -= b;
}

#include"../../math/multiplicative_function.hpp"


void solve(){
    ll n; in(n);
    mint a, b; in(a,b);
    mf_prefix_sum mf(n);
    mint i2 = mint(2).inv();
    auto tbl = mf.table<ar>(
        [&](int p) -> ar {
            return ar{a, b*p};
        },
        [&](ll r) -> ar {
            return ar{a*(r-1), b*(mint(r)*(r+1)*i2-1)};
        },
        [&](int p, ar x) -> ar {
            return ar{x[0], x[1] * p};
        }
    );
    vector<mint> fprime(tbl.size());
    rep(i,tbl.size()){
        fprime[i] = tbl[i][0] + tbl[i][1];
    }
    mint ans = mf.run<mint>(
        [&](int p, int e){
            return a*e + b*p;
        },
        fprime
    );
    out(ans);
}

int main(){
    int t; in(t);
    while (t--){
        solve();
    }
}
#line 1 "test/math/SumofMultiplicativeFunction.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_multiplicative_function"

#line 2 "template/template.hpp"
using namespace std;

#include<bits/stdc++.h>
#line 1 "template/inout_old.hpp"
namespace noya2 {

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;
}

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...);
}

template<typename T>
void out(const vector<vector<T>> &vv){
    int s = (int)vv.size();
    for (int i = 0; i < s; i++) out(vv[i]);
}

struct IoSetup {
    IoSetup(){
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
        cerr << fixed << setprecision(7);
    }
} iosetup_noya2;

} // namespace noya2
#line 1 "template/const.hpp"
namespace noya2{

const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 =  998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";

void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }

} // namespace noya2
#line 2 "template/utils.hpp"

#line 6 "template/utils.hpp"

namespace noya2{

unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
    if (a == 0 || b == 0) return a + b;
    int n = __builtin_ctzll(a); a >>= n;
    int m = __builtin_ctzll(b); b >>= m;
    while (a != b) {
        int mm = __builtin_ctzll(a - b);
        bool f = a > b;
        unsigned long long c = f ? a : b;
        b = f ? b : a;
        a = (c - b) >> mm;
    }
    return a << std::min(n, m);
}

template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(std::abs(a),std::abs(b))); }

long long sqrt_fast(long long n) {
    if (n <= 0) return 0;
    long long x = sqrt(n);
    while ((x + 1) * (x + 1) <= n) x++;
    while (x * x > n) x--;
    return x;
}

template<typename T> T floor_div(const T n, const T d) {
    assert(d != 0);
    return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}

template<typename T> T ceil_div(const T n, const T d) {
    assert(d != 0);
    return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}

template<typename T> void uniq(std::vector<T> &v){
    std::sort(v.begin(),v.end());
    v.erase(unique(v.begin(),v.end()),v.end());
}

template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }

template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }

} // namespace noya2
#line 8 "template/template.hpp"

#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()

using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;

namespace noya2{

/*　~ (. _________ . /)　*/

}

using namespace noya2;


#line 2 "utility/modint.hpp"

#line 4 "utility/modint.hpp"

#line 2 "math/prime.hpp"

#line 4 "math/prime.hpp"
namespace noya2 {

constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

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;
}

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_flag = is_prime_constexpr(n);

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};
    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; 
        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

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_flag = primitive_root_constexpr(m);

} // namespace noya2
#line 6 "utility/modint.hpp"

namespace noya2{

struct barrett {
    unsigned int _m;
    unsigned long long im;
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
    unsigned int umod() const { return _m; }
    unsigned int mul(unsigned int a, unsigned int b) const {
        unsigned long long z = a;
        z *= b;
        unsigned long long x = (unsigned long long)((__uint128_t(z) * im) >> 64);
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

template <int m>
struct static_modint {
    using mint = static_modint;
  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }
    constexpr static_modint() : _v(0) {}
    template<std::signed_integral T>
    constexpr static_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    constexpr static_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    constexpr unsigned int val() const { return _v; }
    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }
    constexpr mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    constexpr mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    constexpr mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (uint)(z % umod());
        return *this;
    }
    constexpr mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint() - *this; }
    constexpr mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    constexpr mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = is_prime_flag<m>;
};


template <int id> struct dynamic_modint {
    using mint = dynamic_modint;
  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template<std::signed_integral T>
    dynamic_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    dynamic_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    uint val() const { return _v; }
    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }
    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }
    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = noya2::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }
    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> noya2::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

template<typename T>
concept Modint = requires (T &a){
    T::mod();
    a.inv();
    a.val();
    a.pow(declval<int>());
};

} // namespace noya2
#line 5 "test/math/SumofMultiplicativeFunction.test.cpp"
using mint = static_modint<469762049>;
using ar = array<mint,2>;
ar &operator+=(ar &a, ar b){
    a[0] += b[0];
    a[1] += b[1];
    return a;
}
ar &operator-=(ar &a, ar b){
    a[0] -= b[0];
    a[1] -= b[1];
    return a;
}
ar operator-(ar a, ar b){
    return a -= b;
}

#line 2 "math/multiplicative_function.hpp"

#line 6 "math/multiplicative_function.hpp"

namespace noya2 {

std::vector<int> prime_enumerate(int N){
    std::vector<bool> psieve(N / 3 + 1, true);
    int qe = psieve.size();
    for (int p = 5, d = 4, i = 1, sqn = std::sqrt(N); p <= sqn; p += d = 6 - d, i++){
        if (!psieve[i]) continue;
        for (int q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p; q < qe; q += r = s - r){
            psieve[q] = false;
        }
    }
    std::vector<int> ret = {2, 3};
    for (int p = 5, d = 4, i = 1; p <= N; p += d = 6 - d, i++){
        if (psieve[i]) ret.emplace_back(p);
    }
    while (!ret.empty() && ret.back() > N) ret.pop_back();
    return ret;
}

struct mf_prefix_sum {
    long long M, sq, s;
    std::vector<int> p;
    int ps;
    mf_prefix_sum(long long m) : M(m) {
        assert(m <= 1e15);
        sq = std::sqrt(M);
        while (sq * sq > M) sq--;
        while ((sq + 1) * (sq + 1) <= M) sq++;
        if (m != 0){
            long long hls = quo(M, sq);
            while (hls != 1 && quo(M, hls - 1) == sq) hls--;
            s = hls + sq;
        }
        p = prime_enumerate(sq);
        ps = p.size();
    }
    // calc : sum[2 <= prime <= M/i] f(prime)
    // T f(int prime) : f(prime)
    // T sum(long long r) : sum[2 <= x <= r] f(x)
    // T mul(int prime, T s) : sum[x in R] f(prime * x), for s = sum[x in R] f(x), R = { x <= M/i | lpf(x) >= prime }
    template<typename T>
    std::vector<T> table(auto f, auto sum, auto mul) const {
        if (M == 0) return {};
        long long hls = s - sq;
        std::vector<T> hl(hls);
        for (int i = 1; i < hls; i++){
            hl[i] = sum(quo(M, i));
        }
        std::vector<T> hs(sq + 1);
        for (int i = 1; i <= sq; i++){
            hs[i] = sum(i);
        }
        T psum = {};
        for (auto &x : p){
            long long x2 = (long long)(x) * x;
            long long imax = std::min<long long>(hls, quo(M, x2) + 1);
            for (long long i = 1, ix = x; i < imax; i++, ix += x){
                hl[i] -= mul(x, (ix < hls ? hl[ix] : hs[quo(M, ix)]) - psum);
            }
            for (int n = sq; n >= x2; n--){
                hs[n] -= mul(x, hs[quo(n, x)] - psum);
            }
            psum += f(x);
        }
        hl.reserve(sq * 2 + 10);
        for (int i = hs.size(); --i; ) hl.push_back(hs[i]);
        assert((int)(hl.size()) == s);
        return hl;
    }
    // calc : sum[1 <= x <= M] f(x), f is multiplicative
    // T f(int prime, int e) : f(prime ^ e)
    template<typename T>
    T run(auto f, const std::vector<T> &Fprime) const {
        if (M == 0) return {};
        assert((int)(Fprime.size()) == s);
        T ans = Fprime[idx(M)] + 1; // + 1 : f(1)
        auto dfs = [&](auto sfs, int i, int c, long long prod, T cur) -> void {
            ans += cur * f(p[i], c + 1);
            long long lim = quo(M, prod);
            if (lim >= 1LL * p[i] * p[i]){
                sfs(sfs, i, c + 1, p[i] * prod, cur);
            }
            cur *= f(p[i], c);
            ans += cur * (Fprime[idx(lim)] - Fprime[idx(p[i])]);
            int j = i + 1;
            for (; j < ps && p[j] < (1 << 21) && 1LL * p[j] * p[j] * p[j] <= lim; j++){
                sfs(sfs, 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(quo(lim, p[j])), id2 = idx(p[j]);
                sm += f(p[j], 1) * (Fprime[id1] - Fprime[id2]);
                ans += cur * sm;
            }
        };
        for (int i = 0; i < ps; i++){
            dfs(dfs, i, 1, p[i], 1);
        }
        return ans;
    }
    long long quo(long long n, long long d) const { return n / d; }
    long long idx(long long n) const { return n <= sq ? s - n : quo(M, n); }
    long long val(long long i) const { return i >= s - sq ? s - i : quo(M, i); }
};

} // namespace noya2
#line 22 "test/math/SumofMultiplicativeFunction.test.cpp"


void solve(){
    ll n; in(n);
    mint a, b; in(a,b);
    mf_prefix_sum mf(n);
    mint i2 = mint(2).inv();
    auto tbl = mf.table<ar>(
        [&](int p) -> ar {
            return ar{a, b*p};
        },
        [&](ll r) -> ar {
            return ar{a*(r-1), b*(mint(r)*(r+1)*i2-1)};
        },
        [&](int p, ar x) -> ar {
            return ar{x[0], x[1] * p};
        }
    );
    vector<mint> fprime(tbl.size());
    rep(i,tbl.size()){
        fprime[i] = tbl[i][0] + tbl[i][1];
    }
    mint ans = mf.run<mint>(
        [&](int p, int e){
            return a*e + b*p;
        },
        fprime
    );
    out(ans);
}

int main(){
    int t; in(t);
    while (t--){
        solve();
    }
}
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