結果
| 問題 |
No.1396 Giri
|
| コンテスト | |
| ユーザー |
 atcoder8
|
| 提出日時 | 2023-04-16 15:20:54 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 145 ms / 2,000 ms |
| コード長 | 22,168 bytes |
| コンパイル時間 | 12,342 ms |
| コンパイル使用メモリ | 380,748 KB |
| 実行使用メモリ | 17,628 KB |
| 最終ジャッジ日時 | 2024-10-12 00:01:52 |
| 合計ジャッジ時間 | 14,741 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 23 |
ソースコード
use atcoder8_library::{
modint::{Modint998244353, Pow},
EratosthenesSieve,
};
type Mint = Modint998244353;
fn main() {
let n = {
let mut line = String::new();
std::io::stdin().read_line(&mut line).unwrap();
line.trim().parse::<usize>().unwrap()
};
let sieve = EratosthenesSieve::new(n);
let mut lcm_factors = vec![0; n + 1];
for i in 1..=n {
let factors = sieve.prime_factorization(i);
for &(p, e) in &factors {
lcm_factors[p] = lcm_factors[p].max(e);
}
}
let mut ans = (1..=n).fold(Mint::new(1), |acc, x| {
acc * Mint::new(x).pow(lcm_factors[x])
});
ans /= (1..=n).rev().find(|&p| sieve.is_prime(p)).unwrap_or(1);
println!("{}", ans.val());
}
pub mod atcoder8_library {
//! 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<usize>,
}
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<usize> = (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 create_divisor_list(&self, n: usize) -> Vec<usize> {
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
}
}
pub mod modint {
use std::ops::{
Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, ShrAssign, Sub, SubAssign,
};
pub trait RemEuclidU32 {
fn rem_euclid_u32(self, modulus: u32) -> u32;
}
/// Calculate the modular multiplicative inverse of `a` with `m` as modulus.
pub fn modinv(a: u32, m: u32) -> u32 {
assert!(m >= 2);
let (mut a, mut b, mut s, mut t) = (a as i64, m as i64, 1, 0);
while b != 0 {
let q = a / b;
a -= q * b;
std::mem::swap(&mut a, &mut b);
s -= q * t;
std::mem::swap(&mut s, &mut t);
}
assert_eq!(
a.abs(),
1,
"The inverse does not exist. gcd(a, m) = {}",
a.abs()
);
s %= m as i64;
if s < 0 {
s += m as i64;
}
s as u32
}
/// This macro implements rem_euclid_u32 for signed integer types of 32 bits or less.
macro_rules! impl_rem_euclid_u32_for_small_signed {
($($small_signed_type:tt),*) => {
$(
impl RemEuclidU32 for $small_signed_type {
fn rem_euclid_u32(self, modulus: u32) -> u32 {
let ret = (self as i32) % (modulus as i32);
if ret >= 0 {
ret as u32
} else {
(ret + modulus as i32) as u32
}
}
}
)*
};
}
/// This macro implements rem_euclid_u32 for 64-bit signed integer types (including isize).
macro_rules! impl_rem_euclid_u32_for_large_signed {
($($large_signed_type:tt),*) => {
$(
impl RemEuclidU32 for $large_signed_type {
fn rem_euclid_u32(self, modulus: u32) -> u32 {
let ret = (self as i64) % (modulus as i64);
if ret >= 0 {
ret as u32
} else {
(ret + modulus as i64) as u32
}
}
}
)*
};
}
/// This macro implements rem_euclid_u32 for unsigned integer types greater than 32 bits.
macro_rules! impl_rem_euclid_u32_for_small_unsigned {
($($small_unsigned_type:tt),*) => {
$(
impl RemEuclidU32 for $small_unsigned_type {
fn rem_euclid_u32(self, modulus: u32) -> u32 {
self as u32 % modulus
}
}
)*
};
}
/// This macro implements rem_euclid_u32 for 64-bit and larger unsigned integer types (including usize).
macro_rules! impl_rem_euclid_u32_for_large_unsigned {
($($large_unsigned_type:tt),*) => {
$(
impl RemEuclidU32 for $large_unsigned_type {
fn rem_euclid_u32(self, modulus: u32) -> u32 {
(self % modulus as $large_unsigned_type) as u32
}
}
)*
};
}
// Implement rem_euclid_u32 for signed integer types of 32 bits or less.
impl_rem_euclid_u32_for_small_signed!(i8, i16, i32);
// Implement rem_euclid_u32 for 64-bit signed integer types (including isize).
impl_rem_euclid_u32_for_large_signed!(i64, isize);
// Implement rem_euclid_u32 for unsigned integer types of 32 bits or more.
impl_rem_euclid_u32_for_small_unsigned!(u8, u16, u32);
// Implement rem_euclid_u32 for unsigned integer types (including usize) of 64 bits or more.
impl_rem_euclid_u32_for_large_unsigned!(u64, u128, usize);
// Implement rem_euclid_u32 for i128.
impl RemEuclidU32 for i128 {
fn rem_euclid_u32(self, modulus: u32) -> u32 {
let ret = self % (modulus as i128);
if ret >= 0 {
ret as u32
} else {
(ret + modulus as i128) as u32
}
}
}
pub trait Pow<T: Copy + ShrAssign> {
fn pow(self, n: T) -> Self;
}
/// Macro to overload binary operation with `$modint_type` for each integer type
macro_rules! impl_ops {
($modint_type:tt, $($other_type:tt),*) => {
$(
impl Add<$other_type> for $modint_type {
type Output = Self;
fn add(self, rhs: $other_type) -> Self::Output {
self + Self::new(rhs)
}
}
impl Add<$modint_type> for $other_type {
type Output = $modint_type;
fn add(self, rhs: $modint_type) -> Self::Output {
$modint_type::new(self) + rhs
}
}
impl Sub<$other_type> for $modint_type {
type Output = Self;
fn sub(self, rhs: $other_type) -> Self::Output {
self - Self::new(rhs)
}
}
impl Sub<$modint_type> for $other_type {
type Output = $modint_type;
fn sub(self, rhs: $modint_type) -> Self::Output {
$modint_type::new(self) - rhs
}
}
impl Mul<$other_type> for $modint_type {
type Output = Self;
fn mul(self, rhs: $other_type) -> Self::Output {
self * Self::new(rhs)
}
}
impl Mul<$modint_type> for $other_type {
type Output = $modint_type;
fn mul(self, rhs: $modint_type) -> Self::Output {
$modint_type::new(self) * rhs
}
}
impl Div<$other_type> for $modint_type {
type Output = Self;
fn div(self, rhs: $other_type) -> Self::Output {
self / Self::new(rhs)
}
}
impl Div<$modint_type> for $other_type {
type Output = $modint_type;
fn div(self, rhs: $modint_type) -> Self::Output {
$modint_type::new(self) / rhs
}
}
impl AddAssign<$other_type> for $modint_type {
fn add_assign(&mut self, other: $other_type) {
*self = *self + Self::new(other);
}
}
impl SubAssign<$other_type> for $modint_type {
fn sub_assign(&mut self, other: $other_type) {
*self = *self - Self::new(other);
}
}
impl MulAssign<$other_type> for $modint_type {
fn mul_assign(&mut self, other: $other_type) {
*self = *self * Self::new(other);
}
}
impl DivAssign<$other_type> for $modint_type {
fn div_assign(&mut self, other: $other_type) {
*self = *self / Self::new(other);
}
}
)*
};
}
/// This macro defines powers of Modint for unsigned integer types.
macro_rules! impl_power_for_unsigned {
($modint_type:tt, $($unsigned_type:tt),*) => {
$(
impl Pow<$unsigned_type> for $modint_type {
fn pow(self, mut n: $unsigned_type) -> Self {
let mut ret = Self::new(1);
let mut mul = self;
while n != 0 {
if n & 1 == 1 {
ret *= mul;
}
mul *= mul;
n >>= 1;
}
ret
}
}
)*
};
}
/// This macro defines powers of Modint for signed integer types of 32 bits or less.
macro_rules! impl_power_for_small_signed {
($modint_type:tt, $($small_signed_type:tt),*) => {
$(
impl Pow<$small_signed_type> for $modint_type {
fn pow(self, n: $small_signed_type) -> Self {
if n >= 0 {
self.pow(n as u32)
} else {
self.pow(-n as u32).inv()
}
}
}
)*
};
}
/// This macro defines the power of Modint for 64-bit signed integer types (including isize).
macro_rules! impl_power_for_large_signed {
($modint_type:tt, $($large_signed_type:tt),*) => {
$(
impl Pow<$large_signed_type> for $modint_type {
fn pow(self, n: $large_signed_type) -> Self {
if n >= 0 {
self.pow(n as u64)
} else {
self.pow(-n as u64).inv()
}
}
}
)*
};
}
/// This macro generates Modint by specifying the type name and modulus.
macro_rules! generate_modint {
($modint_type:tt, $modulus:literal) => {
#[derive(Debug, Default, Hash, Clone, Copy, PartialEq, Eq)]
pub struct $modint_type {
val: u32,
}
impl $modint_type {
const MOD: u32 = $modulus;
}
impl $modint_type {
pub fn new<T: RemEuclidU32>(val: T) -> Self {
Self {
val: val.rem_euclid_u32($modulus),
}
}
pub fn frac<T: RemEuclidU32>(numer: T, denom: T) -> Self {
Self::new(numer) / Self::new(denom)
}
pub fn raw(val: u32) -> Self {
Self { val }
}
pub fn val(&self) -> u32 {
self.val
}
pub fn inv(&self) -> Self {
Self::new(modinv(self.val, $modulus))
}
}
impl<T: RemEuclidU32> From<T> for $modint_type {
fn from(val: T) -> Self {
Self::new(val)
}
}
impl Add for $modint_type {
type Output = Self;
fn add(self, rhs: Self) -> Self::Output {
Self::new(self.val + rhs.val)
}
}
impl Sub for $modint_type {
type Output = Self;
fn sub(self, rhs: Self) -> Self::Output {
Self::new(self.val + $modulus - rhs.val)
}
}
impl Mul for $modint_type {
type Output = Self;
fn mul(self, rhs: Self) -> Self::Output {
Self::new(self.val as u64 * rhs.val as u64)
}
}
impl Div for $modint_type {
type Output = Self;
#[allow(clippy::suspicious_arithmetic_impl)]
fn div(self, rhs: Self) -> Self::Output {
self * rhs.inv()
}
}
impl AddAssign for $modint_type {
fn add_assign(&mut self, other: Self) {
*self = *self + other;
}
}
impl SubAssign for $modint_type {
fn sub_assign(&mut self, other: Self) {
*self = *self - other;
}
}
impl MulAssign for $modint_type {
fn mul_assign(&mut self, other: Self) {
*self = *self * other;
}
}
impl DivAssign for $modint_type {
fn div_assign(&mut self, other: Self) {
*self = *self / other;
}
}
impl Neg for $modint_type {
type Output = Self;
fn neg(self) -> Self::Output {
Self::new(Self::MOD - self.val)
}
}
// Define a binary operation between each integer type and $modint_type.
impl_ops!(
$modint_type,
i8,
i16,
i32,
i64,
i128,
isize,
u8,
u16,
u32,
u64,
u128,
usize
);
// Define powers of Modint for unsigned integer types.
impl_power_for_unsigned!($modint_type, u8, u16, u32, u64, u128, usize);
// Define powers of Modint for signed integer types of 32 bits or less.
impl_power_for_small_signed!($modint_type, i8, i16, i32);
// Define Modint powers for 64-bit signed integer types (including isize).
impl_power_for_large_signed!($modint_type, i64, isize);
// Define the power of Modint for 128-bit signed integer types.
impl Pow<i128> for $modint_type {
fn pow(self, n: i128) -> Self {
if n >= 0 {
self.pow(n as u128)
} else {
self.pow(-n as u128).inv()
}
}
}
};
}
// Define Modint with 998244353 as modulus
generate_modint!(Modint998244353, 998244353);
// Define Modint with 1000000007 as modulus
generate_modint!(Modint1000000007, 1000000007);
}
}