#include "multiplicative-function/gcd-convolution.hpp"
#pragma once #include "divisor-multiple-transform.hpp" template <typename mint> vector<mint> gcd_convolution(const vector<mint>& a, const vector<mint>& b) { assert(a.size() == b.size()); auto s = a, t = b; multiple_transform::zeta_transform(s); multiple_transform::zeta_transform(t); for (int i = 0; i < (int)a.size(); i++) s[i] *= t[i]; multiple_transform::mobius_transform(s); return s; } /** * @brief GCD畳み込み */
#line 2 "multiplicative-function/gcd-convolution.hpp" #line 2 "multiplicative-function/divisor-multiple-transform.hpp" #include <map> #include <vector> using namespace std; #line 2 "prime/prime-enumerate.hpp" // Prime Sieve {2, 3, 5, 7, 11, 13, 17, ...} vector<int> prime_enumerate(int N) { vector<bool> sieve(N / 3 + 1, 1); for (int p = 5, d = 4, i = 1, sqn = sqrt(N); p <= sqn; p += d = 6 - d, i++) { if (!sieve[i]) continue; for (int q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p, qe = sieve.size(); q < qe; q += r = s - r) sieve[q] = 0; } vector<int> ret{2, 3}; for (int p = 5, d = 4, i = 1; p <= N; p += d = 6 - d, i++) if (sieve[i]) ret.push_back(p); while (!ret.empty() && ret.back() > N) ret.pop_back(); return ret; } #line 8 "multiplicative-function/divisor-multiple-transform.hpp" struct divisor_transform { template <typename T> static void zeta_transform(vector<T> &a) { int N = a.size() - 1; auto sieve = prime_enumerate(N); for (auto &p : sieve) for (int k = 1; k * p <= N; ++k) a[k * p] += a[k]; } template <typename T> static void mobius_transform(vector<T> &a) { int N = a.size() - 1; auto sieve = prime_enumerate(N); for (auto &p : sieve) for (int k = N / p; k > 0; --k) a[k * p] -= a[k]; } template <typename I, typename T> static void zeta_transform(map<I, T> &a) { for (auto p = rbegin(a); p != rend(a); p++) for (auto &x : a) { if (p->first == x.first) break; if (p->first % x.first == 0) p->second += x.second; } } template <typename I, typename T> static void mobius_transform(map<I, T> &a) { for (auto &x : a) { for (auto p = rbegin(a); p != rend(a); p++) { if (x.first == p->first) break; if (p->first % x.first == 0) p->second -= x.second; } } } }; struct multiple_transform { template <typename T> static void zeta_transform(vector<T> &a) { int N = a.size() - 1; auto sieve = prime_enumerate(N); for (auto &p : sieve) for (int k = N / p; k > 0; --k) a[k] += a[k * p]; } template <typename T> static void mobius_transform(vector<T> &a) { int N = a.size() - 1; auto sieve = prime_enumerate(N); for (auto &p : sieve) for (int k = 1; k * p <= N; ++k) a[k] -= a[k * p]; } template <typename I, typename T> static void zeta_transform(map<I, T> &a) { for (auto &x : a) for (auto p = rbegin(a); p->first != x.first; p++) if (p->first % x.first == 0) x.second += p->second; } template <typename I, typename T> static void mobius_transform(map<I, T> &a) { for (auto p1 = rbegin(a); p1 != rend(a); p1++) for (auto p2 = rbegin(a); p2 != p1; p2++) if (p2->first % p1->first == 0) p1->second -= p2->second; } }; /** * @brief 倍数変換・約数変換 * @docs docs/multiplicative-function/divisor-multiple-transform.md */ #line 6 "multiplicative-function/gcd-convolution.hpp" template <typename mint> vector<mint> gcd_convolution(const vector<mint>& a, const vector<mint>& b) { assert(a.size() == b.size()); auto s = a, t = b; multiple_transform::zeta_transform(s); multiple_transform::zeta_transform(t); for (int i = 0; i < (int)a.size(); i++) s[i] *= t[i]; multiple_transform::mobius_transform(s); return s; } /** * @brief GCD畳み込み */