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#include "Pollard_factorize.h"
#include "../Prime/Sieve.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/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;
}
// }}}