number/test/arithmetic_function_totient.test.cpp

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• Last update: 2023-12-26 21:26:22+09:00
• Problem: https://judge.yosupo.jp/problem/sum_of_totient_function

Depends on

• Lagrange interpolation（多項式の Lagrange 補間） (formal_power_series/lagrange_interpolation.hpp)
• Sum of exponential times polynomial ($\sum_{i=0}^{N - 1} r^i f(i)$) (formal_power_series/sum_of_exponential_times_polynomial.hpp)
• Sum of exponential times polynomial limit ($\sum_{i=0}^\infty r^i f(i)$) (formal_power_series/sum_of_exponential_times_polynomial_limit.hpp)
• modint.hpp
• Cumulative sum of arithmetic functions （数論的関数の累積和） (number/arithmetic_cumsum.hpp)
• Linear sieve （線形篩） (number/sieve.hpp)

Code

#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_totient_function"
#include "../../modint.hpp"
#include "../arithmetic_cumsum.hpp"
#include <iostream>
using namespace std;

int main() {
    long long N;
    cin >> N;
    cout << euler_phi_cumsum<ModInt<998244353>>(N).sum(N) << '\n';
}

#line 1 "number/test/arithmetic_function_totient.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_totient_function"
#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>

template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }

    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;

    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }

    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }

    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }

    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }

    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;

        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);

        r = std::min(r, n - r);

        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;

        return ret;
    }

    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }

    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }

    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};

using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 3 "formal_power_series/lagrange_interpolation.hpp"

// CUT begin
// Lagrange interpolation
// Input: [f(0), ..., f(N-1)] (length = N), deg(f) < N
// Output: f(x_eval)
// Complexity: O(N)
// Verified: https://atcoder.jp/contests/arc033/tasks/arc033_4
template <typename MODINT> MODINT interpolate_iota(const std::vector<MODINT> ys, MODINT x_eval) {
    const int N = ys.size();
    if (x_eval.val() < N) return ys[x_eval.val()];
    std::vector<MODINT> facinv(N);
    facinv[N - 1] = MODINT(N - 1).fac().inv();
    for (int i = N - 1; i > 0; i--) facinv[i - 1] = facinv[i] * i;
    std::vector<MODINT> numleft(N);
    MODINT numtmp = 1;
    for (int i = 0; i < N; i++) {
        numleft[i] = numtmp;
        numtmp *= x_eval - i;
    }
    numtmp = 1;
    MODINT ret = 0;
    for (int i = N - 1; i >= 0; i--) {
        MODINT tmp = ys[i] * numleft[i] * numtmp * facinv[i] * facinv[N - 1 - i];
        ret += ((N - 1 - i) & 1) ? (-tmp) : tmp;
        numtmp *= x_eval - i;
    }
    return ret;
}
#line 4 "formal_power_series/sum_of_exponential_times_polynomial_limit.hpp"

// CUT begin
// $d$ 次以下の多項式 $f(x)$ と定数 $r$ について，
// $\sum_{i=0}^\infty r^i f(i)$ の値を $[f(0), ..., f(d - 1), f(d)]$ の値から $O(d)$ で計算．
// Requirement: r != 1
// https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit
// Document: https://hitonanode.github.io/cplib-cpp/formal_power_series/sum_of_exponential_times_polynomial_limit.hpp
template <typename MODINT>
MODINT sum_of_exponential_times_polynomial_limit(MODINT r, std::vector<MODINT> init) {
    assert(r != 1);
    if (init.empty()) return 0;
    if (init.size() == 1) return init[0] / (1 - r);
    auto &bs = init;
    const int d = int(bs.size()) - 1;
    MODINT rp = 1;
    for (int i = 1; i <= d; i++) rp *= r, bs[i] = bs[i] * rp + bs[i - 1];
    MODINT ret = 0;
    rp = 1;
    for (int i = 0; i <= d; i++) {
        ret += bs[d - i] * MODINT(d + 1).nCr(i) * rp;
        rp *= -r;
    }
    return ret / MODINT(1 - r).pow(d + 1);
};
#line 6 "formal_power_series/sum_of_exponential_times_polynomial.hpp"

// CUT begin
// $d$ 次以下の多項式 $f(x)$ と定数 $r$ について，
// $\sum_{i=0}^{N-1} r^i f(i)$ の値を $[f(0), ..., f(d - 1), f(d)]$ の値から $O(d)$ で計算．
// Reference:
// - https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial
// - https://min-25.hatenablog.com/entry/2015/04/24/031413
template <typename MODINT>
MODINT sum_of_exponential_times_polynomial(MODINT r, const std::vector<MODINT> &init, long long N) {
    if (N <= 0) return 0;
    if (init.size() == 1) return init[0] * (r == 1 ? MODINT(N) : (1 - r.pow(N)) / (1 - r));

    std::vector<MODINT> S(init.size() + 1);
    MODINT rp = 1;
    for (int i = 0; i < int(init.size()); i++) {
        S[i + 1] = S[i] + init[i] * rp;
        rp *= r;
    }
    if (N < (long long)S.size()) return S[N];

    if (r == 1) return interpolate_iota<MODINT>(S, N);

    const MODINT Sinf = sum_of_exponential_times_polynomial_limit<MODINT>(r, init);
    MODINT rinv = r.inv(), rinvp = 1;
    for (int i = 0; i < int(S.size()); i++) {
        S[i] = (S[i] - Sinf) * rinvp;
        rinvp *= rinv;
    }
    return interpolate_iota<MODINT>(S, N) * r.pow(N) + Sinf;
};
#line 3 "number/sieve.hpp"
#include <map>
#line 5 "number/sieve.hpp"

// CUT begin
// Linear sieve algorithm for fast prime factorization
// Complexity: O(N) time, O(N) space:
// - MAXN = 10^7:  ~44 MB,  80~100 ms (Codeforces / AtCoder GCC, C++17)
// - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17)
// Reference:
// [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers,"
//     Communications of the ACM, 21(12), 999-1003, 1978.
// - https://cp-algorithms.com/algebra/prime-sieve-linear.html
// - https://37zigen.com/linear-sieve/
struct Sieve {
    std::vector<int> min_factor;
    std::vector<int> primes;
    Sieve(int MAXN) : min_factor(MAXN + 1) {
        for (int d = 2; d <= MAXN; d++) {
            if (!min_factor[d]) {
                min_factor[d] = d;
                primes.emplace_back(d);
            }
            for (const auto &p : primes) {
                if (p > min_factor[d] or d * p > MAXN) break;
                min_factor[d * p] = p;
            }
        }
    }
    // Prime factorization for 1 <= x <= MAXN^2
    // Complexity: O(log x)           (x <= MAXN)
    //             O(MAXN / log MAXN) (MAXN < x <= MAXN^2)
    template <class T> std::map<T, int> factorize(T x) const {
        std::map<T, int> ret;
        assert(x > 0 and
               x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1));
        for (const auto &p : primes) {
            if (x < T(min_factor.size())) break;
            while (!(x % p)) x /= p, ret[p]++;
        }
        if (x >= T(min_factor.size())) ret[x]++, x = 1;
        while (x > 1) ret[min_factor[x]]++, x /= min_factor[x];
        return ret;
    }
    // Enumerate divisors of 1 <= x <= MAXN^2
    // Be careful of highly composite numbers https://oeis.org/A002182/list
    // https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)):
    // 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720
    template <class T> std::vector<T> divisors(T x) const {
        std::vector<T> ret{1};
        for (const auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                for (T a = 1, d = 1; d <= p.second; d++) {
                    a *= p.first;
                    ret.push_back(ret[i] * a);
                }
            }
        }
        return ret; // NOT sorted
    }
    // Euler phi functions of divisors of given x
    // Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g
    // Complexity: O(sqrt(x) + d(x))
    template <class T> std::map<T, T> euler_of_divisors(T x) const {
        assert(x >= 1);
        std::map<T, T> ret;
        ret[1] = 1;
        std::vector<T> divs{1};
        for (auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1);
                divs.push_back(divs[i] * p.first);
                for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) {
                    ret[a * p.first] = ret[a] * p.first;
                    divs.push_back(a * p.first);
                }
            }
        }
        return ret;
    }
    // Moebius function Table, (-1)^{# of different prime factors} for square-free x
    // return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683
    std::vector<int> GenerateMoebiusFunctionTable() const {
        std::vector<int> ret(min_factor.size());
        for (unsigned i = 1; i < min_factor.size(); i++) {
            if (i == 1) {
                ret[i] = 1;
            } else if ((i / min_factor[i]) % min_factor[i] == 0) {
                ret[i] = 0;
            } else {
                ret[i] = -ret[i / min_factor[i]];
            }
        }
        return ret;
    }
    // Calculate [0^K, 1^K, ..., nmax^K] in O(nmax)
    // Note: **0^0 == 1**
    template <class MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) const {
        assert(nmax < int(min_factor.size()));
        assert(K >= 0);
        if (K == 0) return std::vector<MODINT>(nmax + 1, 1);
        std::vector<MODINT> ret(nmax + 1);
        ret[0] = 0, ret[1] = 1;
        for (int n = 2; n <= nmax; n++) {
            if (min_factor[n] == n) {
                ret[n] = MODINT(n).pow(K);
            } else {
                ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]];
            }
        }
        return ret;
    }
};
// Sieve sieve((1 << 20));
#line 5 "number/arithmetic_cumsum.hpp"
#include <cmath>
#line 7 "number/arithmetic_cumsum.hpp"

// CUT begin
// Dirichlet product (convolution)
// - a[i] = f(i) (1-origin)
// - A[i - 1] = f(1) + ... + f(i)
// - invA[i - 1] = f(1) + ... + f(N / i)
// Reference:
// https://maspypy.com/dirichlet-%e7%a9%8d%e3%81%a8%e3%80%81%e6%95%b0%e8%ab%96%e9%96%a2%e6%95%b0%e3%81%ae%e7%b4%af%e7%a9%8d%e5%92%8c
template <class T> struct arithmetic_cumsum {
    long long N;
    int K, L;
    bool _verify_shape(const arithmetic_cumsum &x) const {
        return N == x.N and K == x.K and L == x.L;
    }

    std::vector<T> a, A, invA;
    arithmetic_cumsum(long long N_) : N(N_) {
        K = ceill(std::max(sqrtl(N_), powl(N_ / (N_ > 1 ? log2l(N_) : 1.), 2.0 / 3)));
        L = (N_ + K - 1) / K;
        a.resize(K), A.resize(K), invA.resize(L);
    }
    T at(int i) const { // a[i]
        assert(i >= 1 and i <= K);
        return a[i - 1];
    }
    T sum(long long i) const { // a[1] + ... + a[i]
        assert(i >= 1 and i <= N);
        if (i <= K) return A[i - 1];
        long long j = N / i;
        assert(j >= 1 and j <= L);
        return invA[j - 1];
    }

    arithmetic_cumsum operator+(const arithmetic_cumsum &x) const {
        assert(N == x.N and K == x.K and L == x.L);
        auto ret = *this;
        for (int i = 0; i < ret.K; ++i) {
            ret.a[i] += x.a[i];
            ret.A[i] += x.A[i];
        }
        for (int i = 0; i < ret.L; ++i) ret.invA[i] += x.invA[i];
        return ret;
    }

    arithmetic_cumsum operator-(const arithmetic_cumsum &x) const {
        assert(N == x.N and K == x.K and L == x.L);
        auto ret = *this;
        for (int i = 0; i < ret.K; ++i) {
            ret.a[i] -= x.a[i];
            ret.A[i] -= x.A[i];
        }
        for (int i = 0; i < ret.L; ++i) ret.invA[i] -= x.invA[i];
        return ret;
    }
    // Dirichlet product (convolution)
    arithmetic_cumsum operator*(const arithmetic_cumsum &y) const {
        assert(_verify_shape(y));
        const arithmetic_cumsum &x = *this;
        arithmetic_cumsum ret(x.N);
        for (int i = 1; i <= x.K; ++i) {
            for (int j = 1; i * j <= x.K; ++j) ret.a[i * j - 1] += x.at(i) * y.at(j);
        }
        ret.A = ret.a;
        for (int i = 1; i < ret.K; i++) ret.A[i] += ret.A[i - 1];

        for (int l = 1; l <= x.L; ++l) {
            const long long n = x.N / l;
            long long i = 1;
            for (; i * i <= n; ++i) {
                ret.invA[l - 1] += x.at(i) * y.sum(n / i) + x.sum(n / i) * y.at(i);
            }
            ret.invA[l - 1] -= x.sum(i - 1) * y.sum(i - 1);
        }
        return ret;
    }

    bool operator==(const arithmetic_cumsum &y) const {
        return _verify_shape(y) and a == y.a and A == y.A and invA == y.invA;
    }

    arithmetic_cumsum operator/(const arithmetic_cumsum &y) const { // Division (*this) / y
        assert(_verify_shape(y));
        const arithmetic_cumsum &x = *this;
        arithmetic_cumsum ret(x.N);
        assert(y.at(1) != 0);
        const auto y1inv = 1 / y.at(1);
        ret.a = x.a;
        for (int i = 1; i <= x.K; ++i) {
            ret.a[i - 1] *= y1inv;
            for (int j = 2; i * j <= x.K; ++j) ret.a[i * j - 1] -= ret.at(i) * y.at(j);
        }
        ret.A = ret.a;
        for (int i = 2; i <= x.K; ++i) ret.A[i - 1] += ret.A[i - 2];

        for (int l = x.L; l; --l) {
            const long long n = x.N / l;
            if (n <= K) {
                ret.invA[l - 1] = ret.sum(n);
            } else {
                ret.invA[l - 1] = 0;
                long long i = 1;
                T tmp = 0;
                for (; i * i <= n; ++i) {
                    tmp += y.at(i) * ret.sum(n / i) + y.sum(n / i) * ret.at(i);
                }
                tmp -= y.sum(i - 1) * ret.sum(i - 1);
                ret.invA[l - 1] = (x.sum(n) - tmp) * y1inv;
            }
        }
        return ret;
    }

    static arithmetic_cumsum identity(long long n) { // [1, 0, ..., 0]
        arithmetic_cumsum ret(n);
        ret.a[0] = 1;
        ret.A.assign(ret.K, 1);
        ret.invA.assign(ret.L, 1);
        return ret;
    }
};

// zeta(s) = [1, 1, 1, ...]
template <class T> arithmetic_cumsum<T> zeta_cumsum(long long n) {
    arithmetic_cumsum<T> ret(n);
    ret.a.assign(ret.K, 1);
    for (int k = 1; k <= ret.K; k++) ret.A[k - 1] = k;
    for (int l = 1; l <= ret.L; l++) ret.invA[l - 1] = n / l;
    return ret;
}

// zeta(s - 1) = [1, 2, 3, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_1_cumsum(long long n) {
    arithmetic_cumsum<T> ret(n);
    for (int i = 0; i < ret.K; ++i) {
        ret.a[i] = i + 1;
        ret.A[i] = T(i + 1) * (i + 2) / 2;
    }
    for (int l = 1; l <= ret.L; ++l) {
        T u = n / l;
        ret.invA[l - 1] = u * (u + 1) / 2;
    }
    return ret;
}

// zeta(s - k) = [1^k, 2^k, 3^k, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_cumsum(long long n, int k) {
    if (k == 0) return zeta_cumsum<T>(n);
    if (k == 1) return zeta_shift_1_cumsum<T>(n);
    arithmetic_cumsum<T> ret(n);
    auto init_pows = Sieve(k).enumerate_kth_pows<T>(k, k);
    for (int i = 1; i <= ret.K; ++i) {
        ret.a[i - 1] = T(i).pow(k);
        ret.A[i] = ret.a[i] + (i ? ret.A[i - 1] : 0);
    }
    for (int l = 0; l < ret.L; ++l) {
        ret.invA[l] = sum_of_exponential_times_polynomial<T>(1, init_pows, n / (l + 1) + 1);
    }
    return ret;
}

// Euler totient phi function phi(s) = zeta(s - 1) / zeta(s)
template <class T> arithmetic_cumsum<T> euler_phi_cumsum(long long n) {
    return zeta_shift_1_cumsum<T>(n) / zeta_cumsum<T>(n);
}

// 約数関数の累積和
// sigma_k (n) = \sum_{d \mid n} d^k
template <class T> arithmetic_cumsum<T> divisor_func_cumsum(long long n, int k) {
    return zeta_shift_cumsum<T>(n, k) * zeta_cumsum<T>(n);
}

// Cumulative sum of Moebius function (: Mertens function)
template <class T> arithmetic_cumsum<T> moebius_func_cumsum(long long n) {
    return arithmetic_cumsum<T>::identity(n) / zeta_cumsum<T>(n);
}
#line 5 "number/test/arithmetic_function_totient.test.cpp"
using namespace std;

int main() {
    long long N;
    cin >> N;
    cout << euler_phi_cumsum<ModInt<998244353>>(N).sum(N) << '\n';
}

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