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#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" #include "../../template.h" #include "../NumberTheory/cnt_divisors.h" #include "../multiplicative_function.h" const int N = 1000000; MultiplicativeFunction<N + 1> mf; auto divisors = mf.divisors(); #include "../multiplicative_functions_linear.h" void solve() { linear_sieve::linear_sieve_divisors(N + 1); for (int i = 1; i <= N; ++i) { assert(divisors[i] == cnt_divisors(i)); assert(divisors[i] == linear_sieve::cnt_divisors[i]); } cout << "Hello World\n"; }
#line 1 "Math/tests/cnt_divisors_stress.test.cpp" #define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" #line 1 "template.h" #include <bits/stdc++.h> using namespace std; #define FOR(i,a,b) for(int i=(a),_b=(b); i<=_b; i++) #define FORD(i,a,b) for(int i=(a),_b=(b); i>=_b; i--) #define REP(i,a) for(int i=0,_a=(a); i<_a; i++) #define EACH(it,a) for(__typeof(a.begin()) it = a.begin(); it != a.end(); ++it) #define DEBUG(x) { cout << #x << " = "; cout << (x) << endl; } #define PR(a,n) { cout << #a << " = "; FOR(_,1,n) cout << a[_] << ' '; cout << endl; } #define PR0(a,n) { cout << #a << " = "; REP(_,n) cout << a[_] << ' '; cout << endl; } #define sqr(x) ((x) * (x)) // For printing pair, container, etc. // Copied from https://quangloc99.github.io/2021/07/30/my-CP-debugging-template.html template<class U, class V> ostream& operator << (ostream& out, const pair<U, V>& p) { return out << '(' << p.first << ", " << p.second << ')'; } template<class Con, class = decltype(begin(declval<Con>()))> typename enable_if<!is_same<Con, string>::value, ostream&>::type operator << (ostream& out, const Con& con) { out << '{'; for (auto beg = con.begin(), it = beg; it != con.end(); it++) { out << (it == beg ? "" : ", ") << *it; } return out << '}'; } template<size_t i, class T> ostream& print_tuple_utils(ostream& out, const T& tup) { if constexpr(i == tuple_size<T>::value) return out << ")"; else return print_tuple_utils<i + 1, T>(out << (i ? ", " : "(") << get<i>(tup), tup); } template<class ...U> ostream& operator << (ostream& out, const tuple<U...>& t) { return print_tuple_utils<0, tuple<U...>>(out, t); } mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count()); long long get_rand(long long r) { return uniform_int_distribution<long long> (0, r-1)(rng); } template<typename T> vector<T> read_vector(int n) { vector<T> res(n); for (int& x : res) cin >> x; return res; } void solve(); int main() { ios::sync_with_stdio(0); cin.tie(0); solve(); return 0; } #line 1 "Math/NumberTheory/Pollard_factorize.h" // Copied from https://judge.yosupo.jp/submission/61447 // O(N^0.25) // // Tested: // - (up to 10^18; 200 tests) https://judge.yosupo.jp/problem/factorize // - https://oj.vnoi.info/problem/icpc21_beta_l // - https://www.spoj.com/problems/FACT0/ // // Pollard {{{ using ll = long long; using ull = unsigned long long; using ld = long double; ll mult(ll x, ll y, ll md) { ull q = (ld)x * y / md; ll res = ((ull)x * y - q * md); if (res >= md) res -= md; if (res < 0) res += md; return res; } ll powMod(ll x, ll p, ll md) { if (p == 0) return 1; if (p & 1) return mult(x, powMod(x, p - 1, md), md); return powMod(mult(x, x, md), p / 2, md); } bool checkMillerRabin(ll x, ll md, ll s, int k) { x = powMod(x, s, md); if (x == 1) return true; while(k--) { if (x == md - 1) return true; x = mult(x, x, md); if (x == 1) return false; } return false; } bool isPrime(ll x) { if (x == 2 || x == 3 || x == 5 || x == 7) return true; if (x % 2 == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false; if (x < 121) return x > 1; ll s = x - 1; int k = 0; while(s % 2 == 0) { s >>= 1; k++; } if (x < 1LL << 32) { for (ll z : {2, 7, 61}) { if (!checkMillerRabin(z, x, s, k)) return false; } } else { for (ll z : {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) { if (!checkMillerRabin(z, x, s, k)) return false; } } return true; } ll gcd(ll x, ll y) { return y == 0 ? x : gcd(y, x % y); } void pollard(ll x, vector<ll> &ans) { if (isPrime(x)) { ans.push_back(x); return; } ll c = 1; while(true) { c = 1 + get_rand(x - 1); auto f = [&](ll y) { ll res = mult(y, y, x) + c; if (res >= x) res -= x; return res; }; ll y = 2; int B = 100; int len = 1; ll g = 1; while(g == 1) { ll z = y; for (int i = 0; i < len; i++) { z = f(z); } ll zs = -1; int lft = len; while(g == 1 && lft > 0) { zs = z; ll p = 1; for (int i = 0; i < B && i < lft; i++) { p = mult(p, abs(z - y), x); z = f(z); } g = gcd(p, x); lft -= B; } if (g == 1) { y = z; len <<= 1; continue; } if (g == x) { g = 1; z = zs; while(g == 1) { g = gcd(abs(z - y), x); z = f(z); } } if (g == x) break; assert(g != 1); pollard(g, ans); pollard(x / g, ans); return; } } } // return list of all prime factors of x (can have duplicates) vector<ll> factorize(ll x) { vector<ll> ans; for (ll p : {2, 3, 5, 7, 11, 13, 17, 19}) { while(x % p == 0) { x /= p; ans.push_back(p); } } if (x != 1) { pollard(x, ans); } sort(ans.begin(), ans.end()); return ans; } // return pairs of (p, k) where x = product(p^k) vector<pair<ll, int>> factorize_pk(ll x) { auto ps = factorize(x); ll last = -1, cnt = 0; vector<pair<ll, int>> res; for (auto p : ps) { if (p == last) ++cnt; else { if (last > 0) res.emplace_back(last, cnt); last = p; cnt = 1; } } if (cnt > 0) { res.emplace_back(last, cnt); } return res; } vector<ll> get_all_divisors(ll n) { auto pks = factorize_pk(n); vector<ll> res; function<void(int, ll)> gen = [&] (int i, ll prod) { if (i == static_cast<int>(pks.size())) { res.push_back(prod); return; } ll cur_power = 1; for (int cur = 0; cur <= pks[i].second; ++cur) { gen(i+1, prod * cur_power); cur_power *= pks[i].first; } }; gen(0, 1LL); sort(res.begin(), res.end()); return res; } // }}} #line 1 "Math/Prime/Sieve.h" // F is called for each prime // Sieve (odd only + segmented) {{{ template<typename F> void sieve(int MAX, F func) { const int S = sqrt(MAX + 0.5); vector<char> sieve(S + 1, true); vector<array<int, 2>> cp; for (int i = 3; i <= S; i += 2) { if (!sieve[i]) continue; cp.push_back({i, (i * i - 1) / 2}); for (int j = i * i; j <= S; j += 2 * i) sieve[j] = false; } func(2); vector<char> block(S); int high = (MAX - 1) / 2; for (int low = 0; low <= high; low += S) { fill(block.begin(), block.end(), true); for (auto &i : cp) { int p = i[0], idx = i[1]; for (; idx < S; idx += p) block[idx] = false; i[1] = idx - S; } if (low == 0) block[0] = false; for (int i = 0; i < S && low + i <= high; i++) if (block[i]) { func((low + i) * 2 + 1); } }; } // }}} #line 3 "Math/NumberTheory/cnt_divisors.h" // Tested: https://www.spoj.com/problems/NUMDIV/ int64_t cnt_divisors(int64_t n) { assert(n > 0); auto ps = factorize(n); int cnt_ps = ps.size(); int i = 0; int64_t res = 1; while (i < cnt_ps) { int j = i; while (j+1 < cnt_ps && ps[j+1] == ps[j]) ++j; res *= j - i + 2; i = j + 1; } return res; } // Count divisors Using Segmented Sieve O(sieve(sqrt(R)) + (R-L)*log) {{{ // Returns vector of length (r - l + 1), where the i-th element is number of // divisors of i - l vector<int> cnt_divisors_segmented_sieve(int l, int r) { int s = sqrt(r + 0.5); vector<int> primes; auto newPrime = [&] (int p) { primes.push_back(p); }; sieve(s, newPrime); vector<int> cnt(r - l + 1, 1), cur(r - l + 1); std::iota(cur.begin(), cur.end(), l); for (int p : primes) { if (p > r) break; int u = (l + p - 1) / p * p; for (int i = u; i <= r; i += p) { int k = 0; while (cur[i-l] % p == 0) cur[i-l] /= p, ++k; cnt[i - l] *= k + 1; } } for (int i = l; i <= r; ++i) { if (cur[i-l] > 1) cnt[i-l] *= 2; } return cnt; } // }}} #line 1 "Math/multiplicative_function.h" // NOTE: calculate upto N-1 // // Multiplicative function {{{ template<int N> struct MultiplicativeFunction { // Init sieve and pk MultiplicativeFunction() { // Init sieve for (int i = 2; i*i < N; i++) { if (!sieve[i]) { for (int j = i*i; j < N; j += i) { sieve[j] = i; } } } // Init pk for (int i = 2; i < N; i++) { if (!sieve[i]) { pk[i] = {i, 1}; } else { int p = sieve[i]; if (pk[i/p].first == p) { // i = p^k pk[i] = {p, pk[i/p].second + 1}; } else { pk[i] = {-1, 0}; } } } } // Tested: https://cses.fi/problemset/task/1713 array<int, N> divisors() { array<int, N> res; res[1] = 1; for (int i = 2; i < N; i++) { if (pk[i].first > 0) { // i = p^k res[i] = pk[i].second + 1; } else { // i = u * v, gcd(u, v) = 1 int u = i, v = 1; int p = sieve[i]; while (u % p == 0) { u /= p; v *= p; } res[i] = res[u] * res[v]; } } return res; } // mobius(n) = 1 if n is square-free and has *even* number of prime factors // mobius(n) = -1 if n is square-free and has *odd* number of of prime factors // mobius(n) = 0 if n is not square-free array<int, N> mobius() { array<int, N> res; res[1] = 1; for (int i = 2; i < N; ++i) { if (pk[i].first > 0) { // i = p^k res[i] = (pk[i].second >= 2) ? 0 : -1; } else { // i = u * v, gcd(u, v) = 1 int u = i, v = 1; int p = sieve[i]; while (u % p == 0) { u /= p; v *= p; } res[i] = res[u] * res[v]; } } return res; } // private: // sieve[i] == 0 if i is prime, // sieve[i] = any prime factor p otherwise array<int, N> sieve = {0}; // pk[i] = {p, k} if i == p^k // pk[i] = {-1, 0} otherwise array<pair<int,int>, N> pk; }; // }}} #line 6 "Math/tests/cnt_divisors_stress.test.cpp" const int N = 1000000; MultiplicativeFunction<N + 1> mf; auto divisors = mf.divisors(); #line 1 "Math/multiplicative_functions_linear.h" // This is only for calculating multiplicative functions // If we need a fast sieve, see SieveFast.h // From https://codeforces.com/blog/entry/54090 namespace linear_sieve { const int MN = 2e7; vector<int> primes; int smallest_p[MN]; // smallest_p[n] = smallest prime factor of n void linear_sieve_smallest_prime_factor(int n) { primes.clear(); memset(smallest_p, 0, sizeof smallest_p); for (int i = 2; i < n; ++i) { if (!smallest_p[i]) primes.push_back(i); for (int j = 0; j < int(primes.size()) && i * primes[j] < n; ++j) { smallest_p[i * primes[j]] = primes[j]; if (i % primes[j] == 0) break; } } } // Euler Phi {{{ bool is_composite[MN]; int phi[MN]; void linear_sieve_phi(int n) { memset(is_composite, false, sizeof is_composite); primes.clear(); phi[1] = 1; for (int i = 2; i < n; ++i) { if (!is_composite[i]) { primes.push_back(i); phi[i] = i - 1; // i is prime } for (int j = 0; j < (int) primes.size() && i * primes[j] < n; ++j) { is_composite[i * primes[j]] = true; if (i % primes[j] == 0) { phi[i * primes[j]] = phi[i] * primes[j]; //primes[j] divides i break; } else { phi[i * primes[j]] = phi[i] * phi[primes[j]]; //primes[j] does not divide i } } } } // }}} // Number of divisors {{{ int cnt_divisors[MN + 11]; // call linear_sieve_divisors(n+1) to init int cnt[MN + 11]; // power of smallest prime factor of i void linear_sieve_divisors(int n) { // init range [1, n-1] memset(is_composite, false, sizeof is_composite); primes.clear(); cnt_divisors[1] = 1; for (int i = 2; i < n; ++i) { if (!is_composite[i]) { primes.push_back(i); cnt[i] = 1; cnt_divisors[i] = 2; } for (int j = 0; j < (int) primes.size() && i * primes[j] < n; ++j) { int ip = i * primes[j]; is_composite[ip] = true; if (i % primes[j] == 0) { cnt[ip] = cnt[i] + 1; cnt_divisors[ip] = cnt_divisors[i] / (cnt[i] + 1) * (cnt[i] + 2); } else { cnt[ip] = 1; cnt_divisors[ip] = 2 * cnt_divisors[i]; } } } } // }}} } #line 12 "Math/tests/cnt_divisors_stress.test.cpp" void solve() { linear_sieve::linear_sieve_divisors(N + 1); for (int i = 1; i <= N; ++i) { assert(divisors[i] == cnt_divisors(i)); assert(divisors[i] == linear_sieve::cnt_divisors[i]); } cout << "Hello World\n"; }