Library

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:heavy_check_mark: test/math/Sum_of_Totient_Function.test.cpp

Depends on

Code

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

#include "../../math/DirichletSeries.hpp"
#include "../../modint/modint.hpp"
#include "../../template/template.hpp"

namespace ebi {

using mint = modint998244353;

void main_() {
    i64 n;
    std::cin >> n;
    using DirichletSeries = DirichletSeries<mint, 0>;
    DirichletSeries::set_size(n);
    mint ans = (DirichletSeries::zeta1() / DirichletSeries::zeta()).get_sum();
    std::cout << ans << '\n';
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    return 0;
}
#line 1 "test/math/Sum_of_Totient_Function.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_totient_function"

#line 2 "math/DirichletSeries.hpp"

#include <functional>

#include <numeric>

#include <vector>


#line 2 "convolution/dirichlet_convolution.hpp"

#line 4 "convolution/dirichlet_convolution.hpp"

#line 2 "math/eratosthenes_sieve.hpp"

#include <cassert>

#include <cstdint>

#line 6 "math/eratosthenes_sieve.hpp"

/*
    reference: https://37zigen.com/sieve-eratosthenes/
*/

namespace ebi {

struct eratosthenes_sieve {
  private:
    using i64 = std::int_fast64_t;
    int n;
    std::vector<bool> table;

  public:
    eratosthenes_sieve(int _n) : n(_n), table(std::vector<bool>(n + 1, true)) {
        table[1] = false;
        for (i64 i = 2; i * i <= n; i++) {
            if (!table[i]) continue;
            for (i64 j = i; i * j <= n; j++) {
                table[i * j] = false;
            }
        }
    }

    bool is_prime(int p) {
        return table[p];
    }

    std::vector<int> prime_table(int m = -1) {
        if (m < 0) m = n;
        std::vector<int> prime;
        for (int i = 2; i <= m; i++) {
            if (table[i]) prime.emplace_back(i);
        }
        return prime;
    }
};

}  // namespace ebi
#line 2 "math/linear_sieve.hpp"

#line 2 "modint/base.hpp"

#include <concepts>
#include <iostream>
#include <utility>

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 2 "template/int_alias.hpp"

#line 4 "template/int_alias.hpp"

namespace ebi {

using ld = long double;
using std::size_t;
using i8 = std::int8_t;
using u8 = std::uint8_t;
using i16 = std::int16_t;
using u16 = std::uint16_t;
using i32 = std::int32_t;
using u32 = std::uint32_t;
using i64 = std::int64_t;
using u64 = std::uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;

}  // namespace ebi
#line 5 "math/linear_sieve.hpp"

/*
    reference: https://37zigen.com/linear-sieve/
    verify:    https://atcoder.jp/contests/abc162/submissions/25095562
*/

#line 13 "math/linear_sieve.hpp"

namespace ebi {

struct linear_sieve {
  private:
    using u64 = std::uint64_t;
    int n;
    std::vector<int> sieve;
    std::vector<int> prime;

  public:
    linear_sieve(int _n) : n(_n), sieve(std::vector<int>(_n + 1, -1)) {
        for (int i = 2; i <= n; i++) {
            if (sieve[i] < 0) {
                sieve[i] = i;
                prime.emplace_back(i);
            }
            for (auto p : prime) {
                if (u64(p) * u64(i) > u64(n) || p > sieve[i]) break;
                sieve[p * i] = p;
            }
        }
    }

    std::vector<int> prime_table() const {
        return prime;
    }

    std::vector<std::pair<int, int>> prime_power_table(int m) const {
        assert(m <= n);
        std::vector<std::pair<int, int>> table(m + 1, {1, 1});
        for (int i = 2; i <= m; i++) {
            int p = sieve[i];
            table[i] = {p, p};
            if (sieve[i / p] == p) {
                table[i] = table[i / p];
                table[i].second *= p;
            }
        }
        return table;
    }

    std::vector<std::pair<int, int>> factorize(int x) {
        assert(x <= n);
        std::vector<std::pair<int, int>> res;
        while (x > 1) {
            int p = sieve[x];
            int exp = 0;
            if (p < 0) {
                res.emplace_back(x, 1);
                break;
            }
            while (sieve[x] == p) {
                x /= p;
                exp++;
            }
            res.emplace_back(p, exp);
        }
        return res;
    }

    std::vector<int> divisors(int x) {
        assert(x <= n);
        std::vector<int> res;
        res.emplace_back(1);
        auto pf = factorize(x);
        for (auto p : pf) {
            int sz = (int)res.size();
            for (int i = 0; i < sz; i++) {
                int ret = 1;
                for (int j = 0; j < p.second; j++) {
                    ret *= p.first;
                    res.emplace_back(res[i] * ret);
                }
            }
        }
        return res;
    }

    template <class T> std::vector<T> fast_zeta(const std::vector<T> &f) {
        std::vector<T> F = f;
        int sz = f.size();
        assert(sz <= n + 1);
        for (int i = 2; i < sz; i++) {
            if (sieve[i] != i) continue;
            for (int j = (sz - 1) / i; j >= 1; j--) {
                F[j] += F[j * i];
            }
        }
        return F;
    }

    template <class T> std::vector<T> fast_mobius(const std::vector<T> &F) {
        std::vector<T> f = F;
        int sz = F.size();
        assert(sz <= n + 1);
        for (int i = 2; i < sz; i++) {
            if (sieve[i] != i) continue;
            for (int j = 1; j * i < sz; j++) {
                f[j] -= f[j * i];
            }
        }
        return f;
    }

    template <Modint mint> std::vector<mint> pow_table(int m, int k) {
        assert(m <= n && k >= 0);
        std::vector<mint> table(m + 1, 1);
        table[0] = (k == 0);
        for (int i = 2; i <= m; i++) {
            if (sieve[i] == i) {
                table[i] = mint(i).pow(k);
                continue;
            }
            table[i] = table[sieve[i]] * table[i / sieve[i]];
        }
        return table;
    }

    template <Modint mint> std::vector<mint> inv_table() {
        return pow_table(mint::mod() - 2);
    }
};

}  // namespace ebi

#line 7 "convolution/dirichlet_convolution.hpp"

namespace ebi {

template <class T>
std::vector<T> dirichlet_convolution(const std::vector<T> &a,
                                     const std::vector<T> &b) {
    assert(a.size() == b.size());
    int n = a.size() - 1;
    std::vector<T> c(n + 1, 0);
    for (int i = 1; i <= n; i++) {
        for (int j = 1; i * j <= n; j++) {
            c[i * j] += a[i] * b[j];
        }
    }
    return c;
}

template <class T>
std::vector<T> dirichlet_convolution_left_is_multiplicative_function(
    const std::vector<T> &a, const std::vector<T> &b) {
    assert(a.size() == b.size());
    int n = a.size() - 1;
    static int m = 1;
    static std::vector<int> primes;
    if (m < n) {
        while (m < n) m <<= 1;
        eratosthenes_sieve sieve(m);
        primes = sieve.prime_table();
    }
    std::vector<T> c = b;
    for (auto p : primes) {
        if (p > n) break;
        for (int i = n / p; i >= 1; i--) {
            int s = p * i;
            int pk = p, j = i;
            while (1) {
                c[s] += a[pk] * c[j];
                if (j % p != 0) break;
                pk *= p;
                j /= p;
            }
        }
    }
    return c;
}

template <class T>
std::vector<T> dirichlet_convolution_multiplicative_function(
    const std::vector<T> &a, const std::vector<T> &b) {
    assert(a.size() == b.size());
    int n = a.size() - 1;
    static int m = 1;
    static std::vector<std::pair<int, int>> prime_pow_table;
    if (m < n) {
        while (m < n) m <<= 1;
        linear_sieve sieve(m);
        prime_pow_table = sieve.prime_power_table(m);
    }
    std::vector<T> c(n + 1, 0);
    c[1] = a[1] * b[1];
    for (int i = 2; i <= n; i++) {
        auto [p, pk] = prime_pow_table[i];
        if (pk == i) {
            for (int j = 1; j <= i; j *= p) {
                c[i] += a[j] * b[i / j];
            }
        } else {
            c[i] = c[i / pk] * c[pk];
        }
    }
    return c;
}

}  // namespace ebi
#line 9 "math/DirichletSeries.hpp"

namespace ebi {

template <class T, int id> struct DirichletSeries {
  private:
    using Self = DirichletSeries<T, id>;

    void set(std::function<T(i64)> f, std::function<T(i64)> F) {
        for (int i = 1; i <= K; i++) {
            a[i] = f(i);
        }
        for (int i = 1; i <= L; i++) {
            A[i] = F(N / i);
        }
    }

  public:
    DirichletSeries() : a(K + 1), A(L + 1) {}

    DirichletSeries(std::function<T(i64)> f, std::function<T(i64)> F,
                    bool _is_multiplicative = false)
        : a(K + 1), A(L + 1), is_multiplicative(_is_multiplicative) {
        set(f, F);
    }

    Self operator+(const Self &rhs) const noexcept {
        return Self(*this) += rhs;
    }
    Self operator-(const Self &rhs) const noexcept {
        return Self(*this) -= rhs;
    }
    Self operator*(const Self &rhs) const noexcept {
        return Self(*this) *= rhs;
    }
    Self operator/(const Self &rhs) const noexcept {
        return Self(*this) /= rhs;
    }

    Self operator+=(const Self &rhs) noexcept {
        for (int i = 1; i <= K; i++) {
            a[i] += rhs.a[i];
        }
        for (int i = 1; i <= L; i++) {
            A[i] += rhs.A[i];
        }
        return *this;
    }

    Self operator-=(const Self &rhs) noexcept {
        for (int i = 1; i <= K; i++) {
            a[i] -= rhs.a[i];
        }
        for (int i = 1; i <= L; i++) {
            A[i] -= rhs.A[i];
        }
        return *this;
    }

    Self operator*=(const Self &rhs) noexcept {
        Self ret;
        if (is_multiplicative && rhs.is_multiplicative) {
            ret.a = dirichlet_convolution_multiplicative_function(a, rhs.a);
            ret.is_multiplicative = true;
        } else if (is_multiplicative) {
            ret.a =
                dirichlet_convolution_left_is_multiplicative_function(a, rhs.a);
        } else if (rhs.is_multiplicative) {
            ret.a =
                dirichlet_convolution_left_is_multiplicative_function(rhs.a, a);
        } else {
            ret.a = dirichlet_convolution(a, rhs.a);
        }
        std::vector<T> sum_a = a, sum_b = rhs.a;
        for (int i = 1; i < K; i++) {
            sum_a[i + 1] += sum_a[i];
            sum_b[i + 1] += sum_b[i];
        }
        auto get_A = [&](i64 x) -> T {
            if (x <= K) {
                return sum_a[x];
            } else {
                return A[N / x];
            }
        };
        auto get_B = [&](i64 x) -> T {
            if (x <= K) {
                return sum_b[x];
            } else {
                return rhs.A[N / x];
            }
        };
        for (i64 l = L, m = 1; l >= 1; l--) {
            i64 n = N / l;
            while (m * m <= n) m++;
            m--;
            for (int i = 1; i <= m; i++) {
                ret.A[l] +=
                    a[i] * get_B(n / i) + (get_A(n / i) - get_A(m)) * rhs.a[i];
            }
        }
        return ret;
    }

    // c = a / b

    Self operator/=(const Self &b) noexcept {
        Self c = *this;
        T inv_a = b.a[1].inv();
        for (int i = 1; i <= K; i++) {
            c.a[i] *= inv_a;
            for (int j = 2; i * j <= K; j++) {
                c.a[i * j] -= c.a[i] * b.a[j];
            }
        }
        std::vector<T> sum_b = b.a, sum_c = c.a;
        for (int i = 1; i < K; ++i) {
            sum_b[i + 1] += sum_b[i];
            sum_c[i + 1] += sum_c[i];
        }
        auto get_B = [&](i64 x) -> T {
            if (x <= K) {
                return sum_b[x];
            } else {
                return b.A[N / x];
            }
        };
        auto get_C = [&](i64 x) -> T {
            if (x <= K) {
                return sum_c[x];
            } else {
                return c.A[N / x];
            }
        };
        for (i64 l = L, m = 1; l >= 1; l--) {
            i64 n = N / l;
            while (m * m <= n) m++;
            m--;
            for (int i = 2; i <= m; i++) {
                c.A[l] -= b.a[i] * get_C(n / i);
            }

            for (int i = 1; i <= m; i++) {
                c.A[l] -= c.a[i] * (get_B(n / i) - get_B(m));
            }
            c.A[l] *= inv_a;
        }
        return *this = c;
    }

    Self pow(u64 n) const {
        Self res;
        res.a[1] = 1;
        res.is_multiplicative = is_multiplicative;
        std::fill(res.A.begin(), res.A.end(), 1);
        Self x = *this;
        while (n > 0) {
            if (n & 1) res = res * x;
            x = x * x;
            n >>= 1;
        }
        return res;
    }

    T get_sum() {
        return A[1];
    }

    static Self zeta() {
        Self ret;
        std::fill(ret.a.begin(), ret.a.end(), 1);
        for (int i = 1; i <= L; i++) {
            ret.A[i] = N / i;
        }
        ret.is_multiplicative = true;
        return ret;
    }

    static Self zeta1() {
        Self ret;
        ret.is_multiplicative = true;
        std::iota(ret.a.begin(), ret.a.end(), 0);
        T inv2 = T(2).inv();
        for (int i = 1; i <= L; i++) {
            i64 n = N / i;
            ret.A[i] = T(n) * T(n + 1) * inv2;
        }
        return ret;
    }

    static Self zeta2() {
        Self ret;
        ret.is_multiplicative = true;
        for (int i = 1; i <= K; i++) {
            ret.a[i] = i * i;
        }
        T inv6 = T(6).inv();
        for (int i = 1; i <= L; i++) {
            i64 n = N / i;
            ret.A[i] = T(n) * T(n + 1) * T(2 * n + 1) * inv6;
        }
    }

    static void set_size(i64 n) {
        N = n;
        if (N <= 10) {
            K = N;
            L = 1;
        } else if (N <= 5000) {
            K = 1;
            while (K * K < N) K++;
            L = (N + K - 1) / K;
        } else {
            L = 1;
            while (L * L * L / 50 < N) L++;
            K = (N + L - 1) / L;
        }
    }

    static void set_size_multiplicative(i64 n) {
        N = n;
        L = 1;
        while (L * L * L < N) L++;
        K = L * L;
    }

  private:
    static i64 N, K, L;
    static std::vector<std::pair<int, int>> prime_pow_table;
    std::vector<T> a, A;
    bool is_multiplicative = false;
};

template <class T, int id> i64 DirichletSeries<T, id>::N = 1000000;
template <class T, int id> i64 DirichletSeries<T, id>::K = 10000;
template <class T, int id> i64 DirichletSeries<T, id>::L = 100;
template <class T, int id>
std::vector<std::pair<int, int>> DirichletSeries<T, id>::prime_pow_table = {};

}  // namespace ebi
#line 2 "modint/modint.hpp"

#line 5 "modint/modint.hpp"

#line 7 "modint/modint.hpp"

namespace ebi {

template <int m> struct static_modint {
  private:
    using modint = static_modint;

  public:
    static constexpr int mod() {
        return m;
    }

    static constexpr modint raw(int v) {
        modint 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;
    }

    constexpr unsigned int value() const {
        return val();
    }

    constexpr modint &operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    constexpr modint &operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }

    constexpr modint operator++(int) {
        modint res = *this;
        ++*this;
        return res;
    }
    constexpr modint operator--(int) {
        modint res = *this;
        --*this;
        return res;
    }

    constexpr modint &operator+=(const modint &rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    constexpr modint &operator-=(const modint &rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    constexpr modint &operator*=(const modint &rhs) {
        unsigned long long x = _v;
        x *= rhs._v;
        _v = (unsigned int)(x % (unsigned long long)umod());
        return *this;
    }
    constexpr modint &operator/=(const modint &rhs) {
        return *this = *this * rhs.inv();
    }

    constexpr modint operator+() const {
        return *this;
    }
    constexpr modint operator-() const {
        return modint() - *this;
    }

    constexpr modint pow(long long n) const {
        assert(0 <= n);
        modint x = *this, res = 1;
        while (n) {
            if (n & 1) res *= x;
            x *= x;
            n >>= 1;
        }
        return res;
    }
    constexpr modint inv() const {
        assert(_v);
        return pow(umod() - 2);
    }

    friend modint operator+(const modint &lhs, const modint &rhs) {
        return modint(lhs) += rhs;
    }
    friend modint operator-(const modint &lhs, const modint &rhs) {
        return modint(lhs) -= rhs;
    }
    friend modint operator*(const modint &lhs, const modint &rhs) {
        return modint(lhs) *= rhs;
    }

    friend modint operator/(const modint &lhs, const modint &rhs) {
        return modint(lhs) /= rhs;
    }
    friend bool operator==(const modint &lhs, const modint &rhs) {
        return lhs.val() == rhs.val();
    }
    friend bool operator!=(const modint &lhs, const modint &rhs) {
        return !(lhs == rhs);
    }

  private:
    unsigned int _v = 0;

    static constexpr unsigned int umod() {
        return m;
    }
};

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;

}  // namespace ebi
#line 1 "template/template.hpp"
#include <bits/stdc++.h>

#define rep(i, a, n) for (int i = (int)(a); i < (int)(n); i++)
#define rrep(i, a, n) for (int i = ((int)(n)-1); i >= (int)(a); i--)
#define Rep(i, a, n) for (i64 i = (i64)(a); i < (i64)(n); i++)
#define RRep(i, a, n) for (i64 i = ((i64)(n)-i64(1)); i >= (i64)(a); i--)
#define all(v) (v).begin(), (v).end()
#define rall(v) (v).rbegin(), (v).rend()

#line 2 "template/debug_template.hpp"

#line 4 "template/debug_template.hpp"

namespace ebi {

#ifdef LOCAL
#define debug(...)                                                      \
    std::cerr << "LINE: " << __LINE__ << "  [" << #__VA_ARGS__ << "]:", \
        debug_out(__VA_ARGS__)
#else
#define debug(...)
#endif

void debug_out() {
    std::cerr << std::endl;
}

template <typename Head, typename... Tail> void debug_out(Head h, Tail... t) {
    std::cerr << " " << h;
    if (sizeof...(t) > 0) std::cerr << " :";
    debug_out(t...);
}

}  // namespace ebi
#line 2 "template/io.hpp"

#line 5 "template/io.hpp"
#include <optional>
#line 7 "template/io.hpp"

#line 9 "template/io.hpp"

namespace ebi {

template <typename T1, typename T2>
std::ostream &operator<<(std::ostream &os, const std::pair<T1, T2> &pa) {
    return os << pa.first << " " << pa.second;
}

template <typename T1, typename T2>
std::istream &operator>>(std::istream &os, std::pair<T1, T2> &pa) {
    return os >> pa.first >> pa.second;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &vec) {
    for (std::size_t i = 0; i < vec.size(); i++)
        os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    return os;
}

template <typename T>
std::istream &operator>>(std::istream &os, std::vector<T> &vec) {
    for (T &e : vec) std::cin >> e;
    return os;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::optional<T> &opt) {
    if (opt) {
        os << opt.value();
    } else {
        os << "invalid value";
    }
    return os;
}

void fast_io() {
    std::cout << std::fixed << std::setprecision(15);
    std::cin.tie(nullptr);
    std::ios::sync_with_stdio(false);
}

}  // namespace ebi
#line 2 "template/utility.hpp"

#line 5 "template/utility.hpp"

#line 2 "graph/base.hpp"

#line 5 "graph/base.hpp"
#include <ranges>
#line 7 "graph/base.hpp"

#line 2 "data_structure/simple_csr.hpp"

#line 6 "data_structure/simple_csr.hpp"

namespace ebi {

template <class E> struct simple_csr {
    simple_csr() = default;

    simple_csr(int n, const std::vector<std::pair<int, E>>& elements)
        : start(n + 1, 0), elist(elements.size()) {
        for (auto e : elements) {
            start[e.first + 1]++;
        }
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] += start[i];
        }
        auto counter = start;
        for (auto [i, e] : elements) {
            elist[counter[i]++] = e;
        }
    }

    simple_csr(const std::vector<std::vector<E>>& es)
        : start(es.size() + 1, 0) {
        int n = es.size();
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] = (int)es[i].size() + start[i];
        }
        elist.resize(start.back());
        for (auto i : std::views::iota(0, n)) {
            std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]);
        }
    }

    int size() const {
        return (int)start.size() - 1;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }
    auto operator[](int i) {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }

    const auto operator()(int i, int l, int r) const {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }
    auto operator()(int i, int l, int r) {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }

  private:
    std::vector<int> start;
    std::vector<E> elist;
};

}  // namespace ebi
#line 9 "graph/base.hpp"

namespace ebi {

template <class T> struct Edge {
    int from, to;
    T cost;
    int id;
};

template <class E> struct Graph {
    using cost_type = E;
    using edge_type = Edge<cost_type>;

    Graph(int n_) : n(n_) {}

    Graph() = default;

    void add_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        edges.emplace_back(edge_type{u, v, c, m++});
    }

    void add_undirected_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        buff.emplace_back(v, edge_type{v, u, c, m});
        edges.emplace_back(edge_type{u, v, c, m});
        m++;
    }

    void read_tree(int offset = 1, bool is_weighted = false) {
        read_graph(n - 1, offset, false, is_weighted);
    }

    void read_parents(int offset = 1) {
        for (auto i : std::views::iota(1, n)) {
            int p;
            std::cin >> p;
            p -= offset;
            add_undirected_edge(p, i, 1);
        }
        build();
    }

    void read_graph(int e, int offset = 1, bool is_directed = false,
                    bool is_weighted = false) {
        for (int i = 0; i < e; i++) {
            int u, v;
            std::cin >> u >> v;
            u -= offset;
            v -= offset;
            if (is_weighted) {
                cost_type c;
                std::cin >> c;
                if (is_directed) {
                    add_edge(u, v, c);
                } else {
                    add_undirected_edge(u, v, c);
                }
            } else {
                if (is_directed) {
                    add_edge(u, v, 1);
                } else {
                    add_undirected_edge(u, v, 1);
                }
            }
        }
        build();
    }

    void build() {
        assert(!prepared);
        csr = simple_csr<edge_type>(n, buff);
        buff.clear();
        prepared = true;
    }

    int size() const {
        return n;
    }

    int node_number() const {
        return n;
    }

    int edge_number() const {
        return m;
    }

    edge_type get_edge(int i) const {
        assert(prepared);
        return edges[i];
    }

    std::vector<edge_type> get_edges() const {
        assert(prepared);
        return edges;
    }

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        return csr[i];
    }

  private:
    int n, m = 0;

    std::vector<std::pair<int, edge_type>> buff;

    std::vector<edge_type> edges;
    simple_csr<edge_type> csr;
    bool prepared = false;
};

}  // namespace ebi
#line 8 "template/utility.hpp"

namespace ebi {

template <class T> inline bool chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> inline bool chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> T safe_ceil(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return (a / b) + 1;
    else
        return -((-a) / b);
}

template <class T> T safe_floor(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return a / b;
    else
        return -((-a) / b) - 1;
}

constexpr i64 LNF = std::numeric_limits<i64>::max() / 4;

constexpr int INF = std::numeric_limits<int>::max() / 2;

const std::vector<int> dy = {1, 0, -1, 0, 1, 1, -1, -1};
const std::vector<int> dx = {0, 1, 0, -1, 1, -1, 1, -1};

}  // namespace ebi
#line 6 "test/math/Sum_of_Totient_Function.test.cpp"

namespace ebi {

using mint = modint998244353;

void main_() {
    i64 n;
    std::cin >> n;
    using DirichletSeries = DirichletSeries<mint, 0>;
    DirichletSeries::set_size(n);
    mint ans = (DirichletSeries::zeta1() / DirichletSeries::zeta()).get_sum();
    std::cout << ans << '\n';
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    return 0;
}
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