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#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'; }