use proconio::{input, marker::Usize1}; use crate::eratosthenes_sieve::EratosthenesSieve; const MAX: usize = 10_usize.pow(5); fn main() { input! { (n, q): (usize, usize), aa: [usize; n], lrk: [(Usize1, usize, usize); q], } let sieve = EratosthenesSieve::new(MAX); let mut positions_by_divisor = vec![vec![]; MAX + 1]; for (i, &a) in aa.iter().enumerate() { let divisors = sieve.divisors(a); for &divisor in &divisors { positions_by_divisor[divisor].push(i); } } for &(l, r, k) in &lrk { let positions = &positions_by_divisor[k]; let left = positions.partition_point(|&pos| pos < l); let right = positions.partition_point(|&pos| pos < r); println!("{}", right - left); } } pub mod eratosthenes_sieve { //! Implements the Sieve of Eratosthenes. //! //! Finds the smallest prime factor of each number placed on the sieve, //! so it can perform Prime Factorization as well as Primality Test. #[derive(Debug, Clone)] pub struct EratosthenesSieve { sieve: Vec, } impl EratosthenesSieve { /// Constructs a Sieve of Eratosthenes. /// /// # Arguments /// /// * `upper_limit` - The largest number placed on the sieve. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert_eq!(sieve.prime_factorization(12), vec![(2, 2), (3, 1)]); /// ``` pub fn new(upper_limit: usize) -> Self { let mut sieve: Vec = (0..=upper_limit).collect(); for p in (2..).take_while(|&i| i * i <= upper_limit) { if sieve[p] != p { continue; } for i in ((p * p)..=upper_limit).step_by(p) { if sieve[i] == i { sieve[i] = p; } } } Self { sieve } } /// Returns the least prime factor of `n`. /// /// However, if `n` is `1`, then `1` is returned. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert_eq!(sieve.get_least_prime_factor(1), 1); /// assert_eq!(sieve.get_least_prime_factor(2), 2); /// assert_eq!(sieve.get_least_prime_factor(6), 2); /// assert_eq!(sieve.get_least_prime_factor(11), 11); /// assert_eq!(sieve.get_least_prime_factor(27), 3); /// ``` pub fn get_least_prime_factor(&self, n: usize) -> usize { assert_ne!(n, 0, "`n` must be at least 1."); self.sieve[n] } /// Determines if `n` is prime. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert!(!sieve.is_prime(1)); /// assert!(sieve.is_prime(2)); /// assert!(!sieve.is_prime(6)); /// assert!(sieve.is_prime(11)); /// assert!(!sieve.is_prime(27)); /// ``` pub fn is_prime(&self, n: usize) -> bool { n >= 2 && self.sieve[n] == n } /// Performs prime factorization of `n`. /// /// The result of the prime factorization is returned as a /// list of prime factor and exponent pairs. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert_eq!(sieve.prime_factorization(1), vec![]); /// assert_eq!(sieve.prime_factorization(12), vec![(2, 2), (3, 1)]); /// assert_eq!(sieve.prime_factorization(19), vec![(19, 1)]); /// assert_eq!(sieve.prime_factorization(27), vec![(3, 3)]); /// ``` pub fn prime_factorization(&self, n: usize) -> Vec<(usize, usize)> { assert_ne!(n, 0, "`n` must be at least 1."); let mut n = n; let mut factors: Vec<(usize, usize)> = vec![]; while n != 1 { let p = self.sieve[n]; if factors.is_empty() || factors.last().unwrap().0 != p { factors.push((p, 1)); } else { factors.last_mut().unwrap().1 += 1; } n /= p; } factors } /// Creates a list of positive divisors of `n`. /// /// The positive divisors are listed in ascending order. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert_eq!(sieve.create_divisor_list(1), vec![1]); /// assert_eq!(sieve.create_divisor_list(12), vec![1, 2, 3, 4, 6, 12]); /// assert_eq!(sieve.create_divisor_list(19), vec![1, 19]); /// assert_eq!(sieve.create_divisor_list(27), vec![1, 3, 9, 27]); /// ``` pub fn divisors(&self, n: usize) -> Vec { assert_ne!(n, 0, "`n` must be at least 1."); let prime_factors = self.prime_factorization(n); let divisor_num: usize = prime_factors.iter().map(|&(_, exp)| exp + 1).product(); let mut divisors = vec![1]; divisors.reserve(divisor_num - 1); for (p, e) in prime_factors { let mut add_divisors = vec![]; add_divisors.reserve(divisors.len() * e); let mut mul = 1; for _ in 1..=e { mul *= p; for &d in divisors.iter() { add_divisors.push(d * mul); } } divisors.append(&mut add_divisors); } divisors.sort_unstable(); divisors } /// Calculates the number of positive divisors of `n`. /// /// # Examples /// /// ``` /// use atcoder8_library::eratosthenes_sieve::EratosthenesSieve; /// /// let sieve = EratosthenesSieve::new(27); /// assert_eq!(sieve.calc_divisor_num(1), 1); /// assert_eq!(sieve.calc_divisor_num(12), 6); /// assert_eq!(sieve.calc_divisor_num(19), 2); /// assert_eq!(sieve.calc_divisor_num(27), 4); /// ``` pub fn calc_divisor_num(&self, n: usize) -> usize { assert_ne!(n, 0, "`n` must be at least 1."); let mut n = n; let mut divisor_num = 1; let mut cur_p = None; let mut cur_exp = 0; while n != 1 { let p = self.sieve[n]; if Some(p) == cur_p { cur_exp += 1; } else { divisor_num *= cur_exp + 1; cur_p = Some(p); cur_exp = 1; } n /= p; } divisor_num *= cur_exp + 1; divisor_num } } }