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#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 "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 4 "math/linear_sieve.hpp" /* reference: https://37zigen.com/linear-sieve/ verify: https://atcoder.jp/contests/abc162/submissions/25095562 */ #line 12 "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 <class modint> std::vector<modint> pow_table(int k) { std::vector<modint> table(n + 1, 1); table[0] = 0; for (int i = 2; i <= n; i++) { if (sieve[i] == i) { table[i] = modint(i).pow(k); continue; } table[i] = table[sieve[i]] * table[i / sieve[i]]; } return table; } template <class modint> std::vector<modint> inv_table() { return pow_table(modint::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 4 "modint/modint.hpp" #include <iostream> #line 2 "modint/base.hpp" #include <concepts> #line 5 "modint/base.hpp" #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 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) {} constexpr static_modint(long long v) { v %= (long long)umod(); if (v < 0) v += (long long)umod(); _v = (unsigned int)v; } 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" 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) { 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) { 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 { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { 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; }