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#include "math/number-theory/prime-count.hpp"
#include "kth-root-integer.hpp"
#include "prime-table.hpp"
/**
* @brief Prime Count(素数の個数)
*/
template <int64_t LIM = 100000000000LL>
struct PrimeCount {
private:
int64_t sq;
vector<bool> prime;
vector<int64_t> prime_sum, primes;
int64_t p2(int64_t x, int64_t y) {
if (x < 4) return 0;
int64_t a = pi(y);
int64_t b = pi(kth_root_integer(x, 2));
if (a >= b) return 0;
int64_t sum = (a - 2) * (a + 1) / 2 - (b - 2) * (b + 1) / 2;
for (int64_t i = a; i < b; i++) sum += pi(x / primes[i]);
return sum;
}
int64_t phi(int64_t m, int64_t n) {
if (m < 1) return 0;
if (n > m) return 1;
if (n < 1) return m;
if (m <= primes[n - 1] * primes[n - 1]) return pi(m) - n + 1;
if (m <= primes[n - 1] * primes[n - 1] * primes[n - 1] && m <= sq) {
int64_t sx = pi(kth_root_integer(m, 2));
int64_t ans = pi(m) - (sx + n - 2) * (sx - n + 1) / 2;
for (int64_t i = n; i < sx; ++i) ans += pi(m / primes[i]);
return ans;
}
return phi(m, n - 1) - phi(m / primes[n - 1], n - 1);
}
public:
PrimeCount() : sq(kth_root_integer(LIM, 2)), prime_sum(sq + 1) {
prime = prime_table(sq);
for (int i = 1; i <= sq; i++) prime_sum[i] = prime_sum[i - 1] + prime[i];
primes.reserve(prime_sum[sq]);
for (int i = 1; i <= sq; i++)
if (prime[i]) primes.push_back(i);
}
int64_t pi(int64_t n) {
if (n <= sq) return prime_sum[n];
int64_t m = kth_root_integer(n, 3);
int64_t a = pi(m);
return phi(n, a) + a - 1 - p2(n, m);
}
};
#line 1 "math/number-theory/kth-root-integer.hpp"
uint64_t kth_root_integer(uint64_t a, int k) {
if (k == 1) return a;
auto check = [&](uint32_t x) {
uint64_t mul = 1;
for (int j = 0; j < k; j++) {
if (__builtin_mul_overflow(mul, x, &mul)) return false;
}
return mul <= a;
};
uint64_t ret = 0;
for (int i = 31; i >= 0; i--) {
if (check(ret | (1u << i))) ret |= 1u << i;
}
return ret;
}
#line 1 "math/number-theory/prime-table.hpp"
/**
* @brief Prime Table(素数テーブル)
*
*/
vector<bool> prime_table(int n) {
vector<bool> prime(n + 1, true);
if (n >= 0) prime[0] = false;
if (n >= 1) prime[1] = false;
for (int i = 2; i * i <= n; i++) {
if (!prime[i]) continue;
for (int j = i * i; j <= n; j += i) {
prime[j] = false;
}
}
return prime;
}
#line 3 "math/number-theory/prime-count.hpp"
/**
* @brief Prime Count(素数の個数)
*/
template <int64_t LIM = 100000000000LL>
struct PrimeCount {
private:
int64_t sq;
vector<bool> prime;
vector<int64_t> prime_sum, primes;
int64_t p2(int64_t x, int64_t y) {
if (x < 4) return 0;
int64_t a = pi(y);
int64_t b = pi(kth_root_integer(x, 2));
if (a >= b) return 0;
int64_t sum = (a - 2) * (a + 1) / 2 - (b - 2) * (b + 1) / 2;
for (int64_t i = a; i < b; i++) sum += pi(x / primes[i]);
return sum;
}
int64_t phi(int64_t m, int64_t n) {
if (m < 1) return 0;
if (n > m) return 1;
if (n < 1) return m;
if (m <= primes[n - 1] * primes[n - 1]) return pi(m) - n + 1;
if (m <= primes[n - 1] * primes[n - 1] * primes[n - 1] && m <= sq) {
int64_t sx = pi(kth_root_integer(m, 2));
int64_t ans = pi(m) - (sx + n - 2) * (sx - n + 1) / 2;
for (int64_t i = n; i < sx; ++i) ans += pi(m / primes[i]);
return ans;
}
return phi(m, n - 1) - phi(m / primes[n - 1], n - 1);
}
public:
PrimeCount() : sq(kth_root_integer(LIM, 2)), prime_sum(sq + 1) {
prime = prime_table(sq);
for (int i = 1; i <= sq; i++) prime_sum[i] = prime_sum[i - 1] + prime[i];
primes.reserve(prime_sum[sq]);
for (int i = 1; i <= sq; i++)
if (prime[i]) primes.push_back(i);
}
int64_t pi(int64_t n) {
if (n <= sq) return prime_sum[n];
int64_t m = kth_root_integer(n, 3);
int64_t a = pi(m);
return phi(n, a) + a - 1 - p2(n, m);
}
};