# competitive-programming-library

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# upward fast zeta transform on primes (utils/fast_zeta_transform.hpp)

## Code

#pragma once
#include <cassert>
#include <vector>
#include "number/primes.hpp"
#include "utils/macros.hpp"

/**
* @brief upward fast zeta transform on primes
* @note $O(n \log n)$ (or, $O(n \log \log n)$ ???)
* @return $b_i = \sum _ {i \mid j} a_j$
* @note $a_0, b_0$ means the greatest element
*/
template <class CommutativeSemiring>
std::vector<typename CommutativeSemiring::value_type> upward_fast_zeta_transform_on_primes(std::vector<typename CommutativeSemiring::value_type> a, const prepared_primes & primes, const CommutativeSemiring & mon = CommutativeSemiring()) {
assert (a.size() <= primes.size);
if (a.empty()) return a;
for (int64_t p : primes.primes) {
REP3R (x, 1, (a.size() - 1) / p + 1) {
a[x] = mon.mult(a[x], a[p * x]);
}
}
REP3 (x, 1, a.size()) {
a[x] = mon.mult(a[x], a[0]);
}
return a;
}



#line 2 "utils/fast_zeta_transform.hpp"
#include <cassert>
#include <vector>
#line 2 "number/primes.hpp"
#include <algorithm>
#line 4 "number/primes.hpp"
#include <cstdint>
#include <map>
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 8 "number/primes.hpp"

/**
* @note O(\sqrt{n})
*/
struct prepared_primes {
int size;
std::vector<int> sieve;
std::vector<int> primes;

prepared_primes(int size_)
: size(size_) {

sieve.resize(size);
REP3 (p, 2, size) if (sieve[p] == 0) {
primes.push_back(p);
for (int k = p; k < size; k += p) {
if (sieve[k] == 0) {
sieve[k] = p;
}
}
}
}

std::vector<int64_t> list_prime_factors(int64_t n) {
assert (1 <= n and n < (int64_t)size * size);
std::vector<int64_t> result;

// trial division for large part
for (int p : primes) {
if (n < size or n < (int64_t)p * p) {
break;
}
while (n % p == 0) {
n /= p;
result.push_back(p);
}
}

// small part
if (n == 1) {
// nop
} else if (n < size) {
while (n != 1) {
result.push_back(sieve[n]);
n /= sieve[n];
}
} else {
result.push_back(n);
}

assert (std::is_sorted(ALL(result)));
return result;
}

/**
* @note O(1) if n < size; O(sqrt n) if size <= n < size^2
*/
bool is_prime(int64_t n) {
assert (1 <= n and n < (int64_t)size * size);
if (n < size) {
return sieve[n] == n;
}
for (int p : primes) {
if (n < (int64_t)p * p) {
break;
}
if (n % p == 0) {
return false;
}
}
return true;
}

std::vector<int64_t> list_all_factors(int64_t n) {
auto p = list_prime_factors(n);
std::vector<int64_t> d;
d.push_back(1);
for (int l = 0; l < p.size(); ) {
int r = l + 1;
while (r < p.size() and p[r] == p[l]) ++ r;
int n = d.size();
REP (k1, r - l) {
REP (k2, n) {
d.push_back(d[d.size() - n] * p[l]);
}
}
l = r;
}
return d;
}

std::map<int64_t, int> list_prime_factors_as_map(int64_t n) {
std::map<int64_t, int> cnt;
for (int64_t p : list_prime_factors(n)) {
++ cnt[p];
}
return cnt;
}

int64_t euler_totient(int64_t n) {
int64_t phi = 1;
int64_t last = -1;
for (int64_t p : list_prime_factors(n)) {
if (last != p) {
last = p;
phi *= p - 1;
} else {
phi *= p;
}
}
return phi;
}
};
#line 6 "utils/fast_zeta_transform.hpp"

/**
* @brief upward fast zeta transform on primes
* @note $O(n \log n)$ (or, $O(n \log \log n)$ ???)
* @return $b_i = \sum _ {i \mid j} a_j$
* @note $a_0, b_0$ means the greatest element
*/
template <class CommutativeSemiring>
std::vector<typename CommutativeSemiring::value_type> upward_fast_zeta_transform_on_primes(std::vector<typename CommutativeSemiring::value_type> a, const prepared_primes & primes, const CommutativeSemiring & mon = CommutativeSemiring()) {
assert (a.size() <= primes.size);
if (a.empty()) return a;
for (int64_t p : primes.primes) {
REP3R (x, 1, (a.size() - 1) / p + 1) {
a[x] = mon.mult(a[x], a[p * x]);
}
}
REP3 (x, 1, a.size()) {
a[x] = mon.mult(a[x], a[0]);
}
return a;
}