cp-library-cpp
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Util
(library/number/util.hpp)
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- Include:
#include "library/number/util.hpp"
Util
Verified with
- test/src/number/mod_sqrt/dummy.test.cpp
- test/src/number/primitive_root/dummy.test.cpp
- test/src/number/util/abc222_g.test.cpp
Code
#ifndef SUISEN_NUMBER_UTIL
#define SUISEN_NUMBER_UTIL
#include <array>
#include <cassert>
#include <cmath>
#include <numeric>
#include <tuple>
#include <vector>
/**
* @brief Utilities
*/
namespace suisen {
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T powi(T a, int b) {
T res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res *= pow_a;
pow_a *= pow_a;
}
return res;
}
/**
* @brief Calculates the prime factorization of n in O(√n).
* @tparam T integer type
* @param n integer to factorize
* @return vector of { prime, exponent }. It is guaranteed that prime is ascending.
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<std::pair<T, int>> factorize(T n) {
static constexpr std::array primes{ 2, 3, 5, 7, 11, 13 };
static constexpr int next_prime = 17;
static constexpr int size = std::array{ 1, 2, 8, 48, 480, 5760, 92160 } [primes.size() - 1] ;
static constexpr int period = [] {
int res = 1;
for (auto e : primes) res *= e;
return res;
}();
static constexpr struct S : public std::array<int, size> {
constexpr S() {
for (int i = next_prime, j = 0; i < period + next_prime; i += 2) {
bool ok = true;
for (int p : primes) ok &= i % p > 0;
if (ok) (*this)[j++] = i - next_prime;
}
}
} s{};
assert(n > 0);
std::vector<std::pair<T, int>> res;
auto f = [&res, &n](int p) {
if (n % p) return;
int cnt = 0;
do n /= p, ++cnt; while (n % p == 0);
res.emplace_back(p, cnt);
};
for (int p : primes) f(p);
for (T b = next_prime; b * b <= n; b += period) {
for (int offset : s) f(b + offset);
}
if (n != 1) res.emplace_back(n, 1);
return res;
}
/**
* @brief Enumerates divisors of n from its prime-factorized form in O(# of divisors of n) time.
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return vector of divisors (NOT sorted)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> divisors(const std::vector<std::pair<T, int>>& factorized) {
std::vector<T> res{ 1 };
for (auto [p, c] : factorized) {
for (int i = 0, sz = res.size(); i < sz; ++i) {
T d = res[i];
for (int j = 0; j < c; ++j) res.push_back(d *= p);
}
}
return res;
}
/**
* @brief Enumerates divisors of n in O(√n) time.
* @tparam T integer type
* @param n
* @return vector of divisors (NOT sorted)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> divisors(T n) {
return divisors(factorize(n));
}
/**
* @brief Calculates the divisors for i=1,...,n in O(n log n) time.
* @param n upper bound (closed)
* @return 2-dim vector a of length n+1, where a[i] is the vector of divisors of i.
*/
std::vector<std::vector<int>> divisors_table(int n) {
std::vector<std::vector<int>> divs(n + 1);
for (int i = 1; i <= n; ++i) {
for (int j = i; j <= n; j += i) divs[j].push_back(i);
}
return divs;
}
/**
* @brief Calculates φ(n) from its prime-factorized form in O(log n).
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return φ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T totient(const std::vector<std::pair<T, int>>& factorized) {
T res = 1;
for (const auto& [p, c] : factorized) res *= (p - 1) * powi(p, c - 1);
return res;
}
/**
* @brief Calculates φ(n) in O(√n).
* @tparam T integer type
* @param n
* @return φ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T totient(T n) {
return totient(factorize(n));
}
/**
* @brief Calculates φ(i) for i=1,...,n.
* @param n upper bound (closed)
* @return vector a of length n+1, where a[i]=φ(i) for i=1,...,n
*/
std::vector<int> totient_table(int n) {
std::vector<int> res(n + 1);
for (int i = 0; i <= n; ++i) res[i] = (i & 1) == 0 ? i >> 1 : i;
for (int p = 3; p <= n; p += 2) {
if (res[p] != p) continue;
for (int q = p; q <= n; q += p) res[q] /= p, res[q] *= p - 1;
}
return res;
}
/**
* @brief Calculates λ(n) from its prime-factorized form in O(log n).
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return λ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T carmichael(const std::vector<std::pair<T, int>>& factorized) {
T res = 1;
for (const auto& [p, c] : factorized) {
res = std::lcm(res, ((p - 1) * powi(p, c - 1)) >> (p == 2 and c >= 3));
}
return res;
}
/**
* @brief Calculates λ(n) in O(√n).
* @tparam T integer type
* @param n
* @return λ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T carmichael(T n) {
return carmichael(factorize(n));
}
} // namespace suisen
#endif // SUISEN_NUMBER_UTIL#line 1 "library/number/util.hpp"
#include <array>
#include <cassert>
#include <cmath>
#include <numeric>
#include <tuple>
#include <vector>
/**
* @brief Utilities
*/
namespace suisen {
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T powi(T a, int b) {
T res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res *= pow_a;
pow_a *= pow_a;
}
return res;
}
/**
* @brief Calculates the prime factorization of n in O(√n).
* @tparam T integer type
* @param n integer to factorize
* @return vector of { prime, exponent }. It is guaranteed that prime is ascending.
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<std::pair<T, int>> factorize(T n) {
static constexpr std::array primes{ 2, 3, 5, 7, 11, 13 };
static constexpr int next_prime = 17;
static constexpr int size = std::array{ 1, 2, 8, 48, 480, 5760, 92160 } [primes.size() - 1] ;
static constexpr int period = [] {
int res = 1;
for (auto e : primes) res *= e;
return res;
}();
static constexpr struct S : public std::array<int, size> {
constexpr S() {
for (int i = next_prime, j = 0; i < period + next_prime; i += 2) {
bool ok = true;
for (int p : primes) ok &= i % p > 0;
if (ok) (*this)[j++] = i - next_prime;
}
}
} s{};
assert(n > 0);
std::vector<std::pair<T, int>> res;
auto f = [&res, &n](int p) {
if (n % p) return;
int cnt = 0;
do n /= p, ++cnt; while (n % p == 0);
res.emplace_back(p, cnt);
};
for (int p : primes) f(p);
for (T b = next_prime; b * b <= n; b += period) {
for (int offset : s) f(b + offset);
}
if (n != 1) res.emplace_back(n, 1);
return res;
}
/**
* @brief Enumerates divisors of n from its prime-factorized form in O(# of divisors of n) time.
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return vector of divisors (NOT sorted)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> divisors(const std::vector<std::pair<T, int>>& factorized) {
std::vector<T> res{ 1 };
for (auto [p, c] : factorized) {
for (int i = 0, sz = res.size(); i < sz; ++i) {
T d = res[i];
for (int j = 0; j < c; ++j) res.push_back(d *= p);
}
}
return res;
}
/**
* @brief Enumerates divisors of n in O(√n) time.
* @tparam T integer type
* @param n
* @return vector of divisors (NOT sorted)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> divisors(T n) {
return divisors(factorize(n));
}
/**
* @brief Calculates the divisors for i=1,...,n in O(n log n) time.
* @param n upper bound (closed)
* @return 2-dim vector a of length n+1, where a[i] is the vector of divisors of i.
*/
std::vector<std::vector<int>> divisors_table(int n) {
std::vector<std::vector<int>> divs(n + 1);
for (int i = 1; i <= n; ++i) {
for (int j = i; j <= n; j += i) divs[j].push_back(i);
}
return divs;
}
/**
* @brief Calculates φ(n) from its prime-factorized form in O(log n).
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return φ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T totient(const std::vector<std::pair<T, int>>& factorized) {
T res = 1;
for (const auto& [p, c] : factorized) res *= (p - 1) * powi(p, c - 1);
return res;
}
/**
* @brief Calculates φ(n) in O(√n).
* @tparam T integer type
* @param n
* @return φ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T totient(T n) {
return totient(factorize(n));
}
/**
* @brief Calculates φ(i) for i=1,...,n.
* @param n upper bound (closed)
* @return vector a of length n+1, where a[i]=φ(i) for i=1,...,n
*/
std::vector<int> totient_table(int n) {
std::vector<int> res(n + 1);
for (int i = 0; i <= n; ++i) res[i] = (i & 1) == 0 ? i >> 1 : i;
for (int p = 3; p <= n; p += 2) {
if (res[p] != p) continue;
for (int q = p; q <= n; q += p) res[q] /= p, res[q] *= p - 1;
}
return res;
}
/**
* @brief Calculates λ(n) from its prime-factorized form in O(log n).
* @tparam T integer type
* @param factorized a prime-factorized form of n (a vector of { prime, exponent })
* @return λ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T carmichael(const std::vector<std::pair<T, int>>& factorized) {
T res = 1;
for (const auto& [p, c] : factorized) {
res = std::lcm(res, ((p - 1) * powi(p, c - 1)) >> (p == 2 and c >= 3));
}
return res;
}
/**
* @brief Calculates λ(n) in O(√n).
* @tparam T integer type
* @param n
* @return λ(n)
*/
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T carmichael(T n) {
return carmichael(factorize(n));
}
} // namespace suisen