#[allow(unused_imports)] #[cfg(feature = "dbg")] use dbg::lg; use erato::Sieve; type Fp = fp::F998244353; fn main() { let mut buf = ngtio::with_stdin(); let n = buf.u32(); let ans = Sieve::new() .prime_numbers::() .take_while(|&p| p <= n) .map(|p| { let mut n = n; let mut e = 0; let mut m = 0; while n % p == 0 { e += 1; m += 1; n /= p; } while p <= n { m += 1; n /= p; } let pow = if e == m && n == 1 { 2 * (m - 1) } else { m + e }; Fp::new(p).pow(pow as u32) }) .product::(); println!("{}", ans); } // fp {{{ #[allow(dead_code)] mod fp { use std::{ cmp::PartialEq, fmt, hash::{Hash, Hasher}, iter::{successors, Product, Sum}, marker::PhantomData, mem::swap, ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign}, }; pub trait Mod: Clone + Copy + Hash { const P: u32; const K: u32; const R2: u32 = ((1_u128 << 64) % Self::P as u128) as _; // 2 ^ 64 mod P } fn reduce(x: u64) -> u32 { ((x + u64::from(M::K.wrapping_mul(x as u32)) * u64::from(M::P)) >> 32) as u32 } pub fn fact_iter() -> impl Iterator> { (1..).scan(Fp::new(1), |state, x| { let ans = *state; *state *= x; Some(ans) }) } #[allow(clippy::missing_panics_doc)] pub fn fact_build(n: usize) -> [Vec>; 2] { if n == 0 { [Vec::new(), Vec::new()] } else { let fact = fact_iter::().take(n).collect::>(); let mut fact_inv = vec![fact.last().unwrap().recip(); n]; (1..n).rev().for_each(|i| fact_inv[i - 1] = fact_inv[i] * i); [fact, fact_inv] } } pub fn binom_iter() -> impl Iterator>> { successors(Some(vec![Fp::new(1)]), |last| { let mut crr = last.clone(); crr.push(Fp::new(0)); crr[1..].iter_mut().zip(last).for_each(|(x, &y)| *x += y); Some(crr) }) } #[macro_export] macro_rules! define_mod { ($(($Fp: ident, $Mod: ident, $mod: expr, $k: expr),)*) => {$( #[derive(Clone, Debug, Default, Hash, Copy)] pub struct $Mod {} impl Mod for $Mod { const P: u32 = $mod; const K: u32 = $k; } pub type $Fp = Fp<$Mod>; )*} } define_mod! { (F998244353, Mod998244353, 998_244_353, 998_244_351), (F1000000007, Mod1000000007, 1_000_000_007, 2_226_617_417), (F1012924417, Mod1012924417, 1_012_924_417, 1_012_924_415), (F924844033, Mod924844033, 924_844_033, 924_844_031), } #[derive(Clone, Default, Copy)] pub struct Fp { value: u32, __marker: PhantomData, } impl Fp { pub const P: u32 = M::P; pub fn new(value: u32) -> Self { Self::from_raw(reduce::(u64::from(value) * u64::from(M::R2))) } pub fn value(self) -> u32 { let x = reduce::(u64::from(self.value)); if M::P <= x { x - M::P } else { x } } #[allow(clippy::many_single_char_names)] pub fn recip(self) -> Self { assert_ne!(self, Self::new(0), "0 はだめ。"); let mut x = M::P as i32; let mut y = self.value() as i32; let mut u = 0; let mut v = 1; while y != 0 { let q = x / y; x -= q * y; u -= q * v; swap(&mut x, &mut y); swap(&mut u, &mut v); } debug_assert_eq!(x, 1); if u < 0 { debug_assert_eq!(v, M::P as i32); u += v; } Self::new(u as u32) } pub fn pow>(self, exp: T) -> Self { let mut exp = exp.into(); if exp == 0 { return Self::new(1); } let mut base = self; let mut acc = Self::new(1); while 1 < exp { if exp & 1 == 1 { acc *= base; } exp /= 2; base *= base; } acc * base } fn from_raw(value: u32) -> Self { Self { value, __marker: PhantomData, } } } fn simplify(value: i32, p: i32) -> (i32, i32, i32) { if value.abs() < 10_000 { (value, 1, 0) } else { let mut q = p.div_euclid(value); let mut r = p.rem_euclid(value); if value <= 2 * r { q += 1; r -= value; } let (num, pden, ppden) = simplify(r, value); let den = ppden - q * pden; (num, den, pden) } } macro_rules! impl_from_large_int { ($($T: ty), *$(,)?) => {$( impl From<$T> for Fp { fn from(x: $T) -> Self { Self::new(x.rem_euclid(M::P as _) as u32) } } )*} } impl_from_large_int! { u32, u64, u128, usize, i32, i64, i128, isize, } macro_rules! impl_from_small_int { ($($T: ty), *$(,)?) => {$( impl From<$T> for Fp { fn from(x: $T) -> Self { Self::new(x as u32) } } )*} } impl_from_small_int! { u8, u16, i8, i16, } impl PartialEq for Fp { fn eq(&self, other: &Self) -> bool { fn value(fp: Fp) -> u32 { if fp.value >= M::P { fp.value - M::P } else { fp.value } } value(*self) == value(*other) } } impl Eq for Fp {} impl Hash for Fp { fn hash(&self, state: &mut H) { self.value().hash(state); } } impl> AddAssign for Fp { fn add_assign(&mut self, rhs: T) { self.value += rhs.into().value; if M::P * 2 <= self.value { self.value -= M::P * 2; } } } impl> SubAssign for Fp { fn sub_assign(&mut self, rhs: T) { let rhs = rhs.into(); if self.value < rhs.value { self.value += M::P * 2; } self.value -= rhs.value; } } impl> MulAssign for Fp { fn mul_assign(&mut self, rhs: T) { self.value = reduce::(u64::from(self.value) * u64::from(rhs.into().value)); } } #[allow(clippy::suspicious_op_assign_impl)] impl> DivAssign for Fp { fn div_assign(&mut self, rhs: T) { *self *= rhs.into().recip(); } } impl<'a, M: Mod> From<&'a Self> for Fp { fn from(x: &Self) -> Self { *x } } macro_rules! forward_ops { ($(($trait:ident, $method_assign:ident, $method:ident),)*) => {$( impl>> $trait for Fp { type Output = Self; fn $method(mut self, rhs: T) -> Self { self.$method_assign(rhs); self } } impl<'a, M: Mod, T: Into>> $trait for &'a Fp { type Output = Fp; fn $method(self, other: T) -> Self::Output { $trait::$method(*self, other) } } )*}; } forward_ops! { (Add, add_assign, add), (Sub, sub_assign, sub), (Mul, mul_assign, mul), (Div, div_assign, div), } impl Neg for Fp { type Output = Self; fn neg(self) -> Self { Self::from_raw(M::P * 2 - self.value) } } impl Sum for Fp { fn sum>(iter: I) -> Self { iter.fold(Self::new(0), |b, x| b + x) } } impl Product for Fp { fn product>(iter: I) -> Self { iter.fold(Self::new(1), |b, x| b * x) } } impl<'a, M: Mod> Sum<&'a Self> for Fp { fn sum>(iter: I) -> Self { iter.fold(Self::new(0), |b, x| b + x) } } impl<'a, M: Mod> Product<&'a Self> for Fp { fn product>(iter: I) -> Self { iter.fold(Self::new(1), |b, x| b * x) } } impl fmt::Debug for Fp { fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> { let (num, den, _) = simplify(self.value() as i32, M::P as i32); let (num, den) = match den.signum() { 1 => (num, den), -1 => (-num, -den), _ => unreachable!(), }; if den == 1 { write!(f, "{}", num) } else { write!(f, "{}/{}", num, den) } } } impl fmt::Display for Fp { fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result { self.value().fmt(f) } } } // }}} // erato {{{ #[allow(dead_code)] mod erato { mod converters { use { super::{Int, PrimeFactorsByLookup, PrimeFactorsByTrialDivision}, std::{iter::Peekable, marker::PhantomData}, }; pub trait PrimeFactors: Sized + Iterator { fn unique(self) -> Unique { Unique { iter: self, prev: None, } } fn rle(self) -> Rle { Rle { iter: self.peekable(), _marker: PhantomData, } } } impl<'a, T: Int> PrimeFactors for PrimeFactorsByTrialDivision<'a, T> {} impl<'a, T: Int> PrimeFactors for PrimeFactorsByLookup<'a, T> {} pub struct Unique> { iter: P, prev: Option, } impl> Iterator for Unique { type Item = P::Item; fn next(&mut self) -> Option { let prev = self.prev; let res = self.iter.find(|&p| Some(p) != prev); self.prev = res; res } } pub struct Rle> { iter: Peekable

, _marker: PhantomData, } impl<'a, T: Int, P: PrimeFactors> Iterator for Rle { type Item = (P::Item, usize); fn next(&mut self) -> Option { if let Some(p) = self.iter.next() { let mut multi = 1; while self.iter.peek() == Some(&p) { multi += 1; self.iter.next(); } Some((p, multi)) } else { None } } } } mod int { use std::{ fmt::Debug, ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Rem, RemAssign, Sub, SubAssign}, }; pub trait Int: Debug + Copy + Ord + Add + AddAssign + Sub + SubAssign + Mul + MulAssign + Div + DivAssign + Rem + RemAssign { fn zero() -> Self; fn one() -> Self; fn two() -> Self; fn as_usize(self) -> usize; fn from_usize(src: usize) -> Self; } macro_rules! impl_int { ($($t:ty),* $(,)?) => {$( impl Int for $t { fn zero() -> Self { 0 } fn one() -> Self { 1 } fn two() -> Self { 2 } fn as_usize(self) -> usize { self as usize } fn from_usize(src: usize) -> Self { src as Self } } )*} } impl_int! { usize, u8, u16, u32, u64, u128, isize, i8, i16, i32, i64, i128, } } mod lpd_sieve { use super::{ sieve_base::{PrimeFactorsByLookup, PrimeNumbers}, sieve_kind, Int, SieveBase, }; #[derive(Default, Debug, Clone, PartialEq)] pub struct LpdSieve { base: SieveBase, } impl LpdSieve { pub fn new() -> Self { Self { base: SieveBase::new(), } } pub fn is_empty(&self) -> bool { self.base.is_empty() } pub fn len(&self) -> usize { self.base.len() } pub fn with_len(n: usize) -> Self { Self { base: SieveBase::with_len(n), } } pub fn is_prime(&mut self, x: T) -> bool { self.base.is_prime(x) } pub fn lpd(&mut self, x: T) -> T { self.base.lpd(x) } pub fn prime_numbers(&mut self) -> PrimeNumbers { self.base.prime_numbers() } pub fn prime_factors(&mut self, n: T) -> PrimeFactorsByLookup { self.base.prime_factors_by_lookup(n) } } } mod sieve { use super::{ sieve_base::{PrimeFactorsByTrialDivision, PrimeNumbers}, sieve_kind, Int, SieveBase, }; #[derive(Default, Debug, Clone, PartialEq)] pub struct Sieve { base: SieveBase, } impl Sieve { pub fn new() -> Self { Self { base: SieveBase::new(), } } pub fn is_empty(&self) -> bool { self.base.is_empty() } pub fn len(&self) -> usize { self.base.len() } pub fn with_len(n: usize) -> Self { Self { base: SieveBase::with_len(n), } } pub fn is_prime(&mut self, x: T) -> bool { self.base.is_prime(x) } pub fn prime_numbers(&mut self) -> PrimeNumbers { self.base.prime_numbers() } pub fn prime_factors(&mut self, n: T) -> PrimeFactorsByTrialDivision { self.base.prime_factors_by_trial_division(n) } } } mod sieve_base { use { super::{ sieve_kind::{self, SieveKind}, Int, PrimeFactors, Rle, Unique, }, std::marker::PhantomData, }; #[derive(Debug, Clone, PartialEq)] pub struct SieveBase { sieve: Vec, list: Vec, } impl SieveBase { pub fn new() -> Self { Self { sieve: S::new(), list: Vec::new(), } } pub fn is_empty(&self) -> bool { self.sieve.is_empty() } pub fn len(&self) -> usize { self.sieve.len() } pub fn with_len(n: usize) -> Self { let sieve = S::construct(n); let list = sieve .iter() .enumerate() .filter(|&(index, &b)| S::is_prime(index, b)) .map(|(index, _)| index) .collect(); Self { sieve, list } } pub fn is_prime(&mut self, x: T) -> bool { assert!(T::zero() <= x); let x = x.as_usize(); if self.sieve.len() <= x { *self = Self::with_len(x + 1); } S::is_prime(x, self.sieve[x.as_usize()]) } pub fn prime_numbers(&mut self) -> PrimeNumbers { PrimeNumbers { sieve: self, index: 0, _marker: PhantomData, } } fn extend(&mut self, len: usize) { assert!(2 * self.len() <= len); *self = Self::with_len(len); } } impl Default for SieveBase { fn default() -> Self { Self::new() } } impl SieveBase { pub fn prime_factors_by_trial_division( &mut self, n: T, ) -> PrimeFactorsByTrialDivision { assert!(T::zero() < n); let mut prime_numbers = self.prime_numbers(); PrimeFactorsByTrialDivision { p: prime_numbers.next().unwrap(), prime_numbers, n, } } } pub struct PrimeNumbers<'a, S: SieveKind, T: Int> { sieve: &'a mut SieveBase, index: usize, _marker: PhantomData, } pub struct PrimeFactorsByTrialDivision<'a, T: Int> { prime_numbers: PrimeNumbers<'a, sieve_kind::Boolean, T>, p: T, n: T, } impl<'a, S: SieveKind, T: Int> Iterator for PrimeNumbers<'a, S, T> { type Item = T; fn next(&mut self) -> Option { let Self { sieve, index, .. } = self; let p = if let Some(&p) = sieve.list.get(*index) { T::from_usize(p) } else { sieve.extend((sieve.len() * 2).max(3)); T::from_usize(sieve.list[*index]) }; *index += 1; Some(p) } } impl PrimeFactorsByTrialDivision<'_, T> { pub fn unique(self) -> Unique { PrimeFactors::unique(self) } pub fn rle(self) -> Rle { PrimeFactors::rle(self) } } impl<'a, T: Int> Iterator for PrimeFactorsByTrialDivision<'a, T> { type Item = T; fn next(&mut self) -> Option { let Self { prime_numbers, p, n, } = self; if *n == T::one() { None } else { while *n % *p != T::zero() { if *n <= *p * *p { *p = *n; break; } *p = prime_numbers.next().unwrap(); } *n /= *p; Some(*p) } } } pub struct PrimeFactorsByLookup<'a, T: Int> { sieve: &'a mut SieveBase, n: T, } impl SieveBase { pub fn prime_factors_by_lookup(&mut self, n: T) -> PrimeFactorsByLookup { assert!(T::zero() < n); PrimeFactorsByLookup { sieve: self, n } } pub fn lpd(&mut self, n: T) -> T { let n = n.as_usize(); if self.sieve.len() <= n { self.extend(2 * (n + 1)); } T::from_usize(self.sieve[n]) } } impl PrimeFactorsByLookup<'_, T> { pub fn unique(self) -> Unique { PrimeFactors::unique(self) } pub fn rle(self) -> Rle { PrimeFactors::rle(self) } } impl<'a, T: Int> Iterator for PrimeFactorsByLookup<'a, T> { type Item = T; fn next(&mut self) -> Option { let Self { sieve, n } = self; if *n == T::one() { None } else { let p = sieve.lpd(*n); *n /= p; Some(p) } } } } mod sieve_kind { pub trait SieveKind { type SieveValue: Copy; fn new() -> Vec; fn construct(len: usize) -> Vec; fn is_prime(index: usize, b: Self::SieveValue) -> bool; } #[derive(Debug, Clone, Copy, PartialEq)] pub enum Boolean {} #[derive(Debug, Clone, Copy, PartialEq)] pub enum Usize {} impl SieveKind for Boolean { type SieveValue = bool; fn new() -> Vec { Vec::new() } fn construct(len: usize) -> Vec { construct_is_prime_table(len) } fn is_prime(_index: usize, b: Self::SieveValue) -> bool { b } } impl SieveKind for Usize { type SieveValue = usize; fn new() -> Vec { Vec::new() } fn construct(len: usize) -> Vec { construct_lpd_table(len) } fn is_prime(index: usize, b: Self::SieveValue) -> bool { index == b } } pub fn construct_is_prime_table(n: usize) -> Vec { let mut is_prime = vec![true; n]; (0..2.min(n)).for_each(|i| is_prime[i] = false); for p in (2..).take_while(|&p| p * p < n) { if !is_prime[p] { continue; } let mut i = p * p; while i < n { is_prime[i] = false; i += p; } } is_prime } fn construct_lpd_table(n: usize) -> Vec { let mut lpd = vec![std::usize::MAX; n]; for p in 2..n { if lpd[p] != std::usize::MAX { continue; } lpd[p] = p; let mut i = p * p; while i < n { if lpd[i] == std::usize::MAX { lpd[i] = p; } i += p; } } lpd } } use sieve_base::SieveBase; pub use { converters::{PrimeFactors, Rle, Unique}, int::Int, lpd_sieve::LpdSieve, sieve::Sieve, sieve_base::{PrimeFactorsByLookup, PrimeFactorsByTrialDivision, PrimeNumbers}, }; } // }}} // template {{{ #[cfg(not(feature = "dbg"))] #[allow(unused_macros)] #[macro_export] macro_rules! lg { ($($expr:expr),*) => {}; } #[allow(dead_code)] mod ngtio { mod i { use std::{ io::{self, BufRead}, iter, }; pub use self::{ multi_token::{Leaf, Parser, ParserTuple, RawTuple, Tuple, VecLen}, token::{Token, Usize1}, }; pub fn with_stdin() -> Tokenizer> { io::BufReader::new(io::stdin()).tokenizer() } pub fn with_str(src: &str) -> Tokenizer<&[u8]> { src.as_bytes().tokenizer() } pub struct Tokenizer { queue: Vec, // FIXME: String のみにすると速そうです。 scanner: S, } macro_rules! prim_method { ($name:ident: $T:ty) => { pub fn $name(&mut self) -> $T { <$T>::leaf().parse(self) } }; ($name:ident) => { prim_method!($name: $name); }; } macro_rules! prim_methods { ($name:ident: $T:ty; $($rest:tt)*) => { prim_method!($name:$T); prim_methods!($($rest)*); }; ($name:ident; $($rest:tt)*) => { prim_method!($name); prim_methods!($($rest)*); }; () => () } impl Tokenizer { pub fn token(&mut self) -> String { self.load(); self.queue.pop().expect("入力が終了したのですが。") } pub fn new(scanner: S) -> Self { Self { queue: Vec::new(), scanner, } } fn load(&mut self) { while self.queue.is_empty() { let mut s = String::new(); let length = self.scanner.read_line(&mut s).unwrap(); // 入力が UTF-8 でないときにエラーだそうです。 if length == 0 { break; } self.queue = s.split_whitespace().rev().map(str::to_owned).collect(); } } pub fn skip_line(&mut self) { assert!( self.queue.is_empty(), "行の途中で呼ばないでいただきたいです。現在のトークンキュー: {:?}", &self.queue ); self.load(); } pub fn end(&mut self) { self.load(); assert!(self.queue.is_empty(), "入力はまだあります!"); } pub fn parse(&mut self) -> T::Output { T::parse(&self.token()) } pub fn parse_collect(&mut self, n: usize) -> B where B: iter::FromIterator, { iter::repeat_with(|| self.parse::()).take(n).collect() } pub fn tuple(&mut self) -> ::Output { T::leaf_tuple().parse(self) } pub fn vec(&mut self, len: usize) -> Vec { T::leaf().vec(len).parse(self) } pub fn vec_tuple( &mut self, len: usize, ) -> Vec<::Output> { T::leaf_tuple().vec(len).parse(self) } pub fn vec2(&mut self, height: usize, width: usize) -> Vec> { T::leaf().vec(width).vec(height).parse(self) } pub fn vec2_tuple( &mut self, height: usize, width: usize, ) -> Vec::Output>> where T: RawTuple, { T::leaf_tuple().vec(width).vec(height).parse(self) } prim_methods! { u8; u16; u32; u64; u128; usize; i8; i16; i32; i64; i128; isize; f32; f64; char; string: String; } } mod token { use super::multi_token::Leaf; use std::{any, fmt, marker, str}; pub trait Token: Sized { type Output; fn parse(s: &str) -> Self::Output; fn leaf() -> Leaf { Leaf(marker::PhantomData) } } impl Token for T where T: str::FromStr, ::Err: fmt::Debug, { type Output = T; fn parse(s: &str) -> Self::Output { s.parse().unwrap_or_else(|_| { panic!("Parse error!: ({}: {})", s, any::type_name::(),) }) } } pub struct Usize1 {} impl Token for Usize1 { type Output = usize; fn parse(s: &str) -> Self::Output { usize::parse(s) .checked_sub(1) .expect("Parse error! (Zero substruction error of Usize1)") } } } mod multi_token { use super::{Token, Tokenizer}; use std::{io::BufRead, iter, marker}; pub trait Parser: Sized { type Output; fn parse(&self, server: &mut Tokenizer) -> Self::Output; fn vec(self, len: usize) -> VecLen { VecLen { len, elem: self } } } pub struct Leaf(pub(super) marker::PhantomData); impl Parser for Leaf { type Output = T::Output; fn parse(&self, server: &mut Tokenizer) -> T::Output { server.parse::() } } pub struct VecLen { pub len: usize, pub elem: T, } impl Parser for VecLen { type Output = Vec; fn parse(&self, server: &mut Tokenizer) -> Self::Output { iter::repeat_with(|| self.elem.parse(server)) .take(self.len) .collect() } } pub trait RawTuple { type LeafTuple: Parser; fn leaf_tuple() -> Self::LeafTuple; } pub trait ParserTuple { type Tuple: Parser; fn tuple(self) -> Self::Tuple; } pub struct Tuple(pub T); macro_rules! impl_tuple { ($($t:ident: $T:ident),*) => { impl<$($T),*> Parser for Tuple<($($T,)*)> where $($T: Parser,)* { type Output = ($($T::Output,)*); #[allow(unused_variables)] fn parse(&self, server: &mut Tokenizer) -> Self::Output { match self { Tuple(($($t,)*)) => { ($($t.parse(server),)*) } } } } impl<$($T: Token),*> RawTuple for ($($T,)*) { type LeafTuple = Tuple<($(Leaf<$T>,)*)>; fn leaf_tuple() -> Self::LeafTuple { Tuple(($($T::leaf(),)*)) } } impl<$($T: Parser),*> ParserTuple for ($($T,)*) { type Tuple = Tuple<($($T,)*)>; fn tuple(self) -> Self::Tuple { Tuple(self) } } }; } impl_tuple!(); impl_tuple!(t1: T1); impl_tuple!(t1: T1, t2: T2); impl_tuple!(t1: T1, t2: T2, t3: T3); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6); impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6, t7: T7); impl_tuple!( t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6, t7: T7, t8: T8 ); } trait Scanner: BufRead + Sized { fn tokenizer(self) -> Tokenizer { Tokenizer::new(self) } } impl Scanner for R {} } pub use self::i::{with_stdin, with_str}; pub mod prelude { pub use super::i::{Parser, ParserTuple, RawTuple, Token, Usize1}; } } // }}}