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#include "nt/mertens.hpp"
#include "nt/array_on_floor.hpp" #include "nt/mobius_table.hpp" #include "enumerate/floor_range.hpp" template <typename T> struct Mertens { Array_On_Floor<T> sum; Mertens() {} Mertens(u64 N, u64 K = -1) { build(N, K); } void build(u64 N, u64 K = -1) { sum = Array_On_Floor<T>(N); if (K == u64(-1)) { K = pow(N, 0.67); } vc<T> A = mobius_table<T>(K); FOR(k, 1, K) A[k + 1] += A[k]; FOR(i, len(sum)) { u64 n = sum.get_floor(i); if (n <= K) { sum.dat[i] = A[n]; continue; } T ans = 1; floor_range(n, [&](u64 q, u64 l, u64 r) -> void { if (q == n) return; ans -= sum[q] * T(r - l); }); sum.dat[i] = ans; } } T operator[](u64 n) { return sum[n]; } };
#line 1 "nt/array_on_floor.hpp" // N=10 だと dat = {dp[1], dp[2], dp[3], dp[5], dp[10]} みたいになる // hashmap より数倍高速 template <typename T> struct Array_On_Floor { u64 N; u32 n, sq; vc<T> dat; Array_On_Floor() {} Array_On_Floor(u64 N, T default_value = T{}) : N(N) { assert(N <= u64(1) << 50); sq = sqrtl(N); n = (u64(sq) * sq + sq <= N ? sq : sq - 1); dat.resize(n + sq, default_value); } u32 size() { return dat.size(); } T& operator[](u64 d) { int i = get_index(d); return dat[i]; } inline u32 get_index(u64 d) { assert(d > 0); if (d <= n) return d - 1; return dat.size() - u32(double(N) / d); } // dat[i] に対応する floor u64 get_floor(u32 i) { return (i < n ? 1 + i : double(N) / (n + sq - i)); } template <typename F> void enumerate_all(F f) { FOR(i, len(dat)) { f(get_floor(i), dat[i]); } } }; #line 2 "nt/primetable.hpp" template <typename T = int> vc<T> primetable(int LIM) { ++LIM; const int S = 32768; static int done = 2; static vc<T> primes = {2}, sieve(S + 1); if (done < LIM) { done = LIM; primes = {2}, sieve.assign(S + 1, 0); const int R = LIM / 2; primes.reserve(int(LIM / log(LIM) * 1.1)); vc<pair<int, int>> cp; for (int i = 3; i <= S; i += 2) { if (!sieve[i]) { cp.eb(i, i * i / 2); for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1; } } for (int L = 1; L <= R; L += S) { array<bool, S> block{}; for (auto& [p, idx]: cp) for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1; FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1); } } int k = LB(primes, LIM + 1); return {primes.begin(), primes.begin() + k}; } #line 3 "nt/zeta.hpp" template <typename T> void divisor_zeta(vc<T>& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3(x, 1, N / p + 1) A[p * x] += A[x]; } } template <typename T> void divisor_mobius(vc<T>& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3_R(x, 1, N / p + 1) A[p * x] -= A[x]; } } template <typename T> void multiplier_zeta(vc<T>& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3_R(x, 1, N / p + 1) A[x] += A[p * x]; } } template <typename T> void multiplier_mobius(vc<T>& A) { assert(A[0] == 0); int N = len(A) - 1; auto P = primetable(N); for (auto&& p: P) { FOR3(x, 1, N / p + 1) A[x] -= A[p * x]; } } #line 2 "nt/mobius_table.hpp" template<typename T> vc<T> mobius_table(int N){ vc<T> mu(N + 1); mu[1] = T(1); divisor_mobius(mu); return mu; } #line 1 "enumerate/floor_range.hpp" // 商が q の区間 [l,r) を q について昇順 template <typename F> void floor_range(u64 N, F f) { assert(N <= (u64(1) << 50)); u64 sq = sqrtl(N); u32 n = (sq * sq + sq <= N ? sq : sq - 1); u64 prev = N + 1; for (u32 q = 1; q <= n; ++q) { u64 x = double(N) / (q + 1) + 1; f(q, x, prev), prev = x; } for (u32 l = sq; l >= 1; --l) { f(u64(double(N) / l), l, l + 1); } } #line 4 "nt/mertens.hpp" template <typename T> struct Mertens { Array_On_Floor<T> sum; Mertens() {} Mertens(u64 N, u64 K = -1) { build(N, K); } void build(u64 N, u64 K = -1) { sum = Array_On_Floor<T>(N); if (K == u64(-1)) { K = pow(N, 0.67); } vc<T> A = mobius_table<T>(K); FOR(k, 1, K) A[k + 1] += A[k]; FOR(i, len(sum)) { u64 n = sum.get_floor(i); if (n <= K) { sum.dat[i] = A[n]; continue; } T ans = 1; floor_range(n, [&](u64 q, u64 l, u64 r) -> void { if (q == n) return; ans -= sum[q] * T(r - l); }); sum.dat[i] = ans; } } T operator[](u64 n) { return sum[n]; } };