結果

問題 No.1760 Setwise Coprime
ユーザー akakimidoriakakimidori
提出日時 2022-06-11 11:30:07
言語 Rust
(1.77.0)
結果
AC  
実行時間 17 ms / 2,000 ms
コード長 10,461 bytes
コンパイル時間 1,650 ms
コンパイル使用メモリ 181,784 KB
実行使用メモリ 4,348 KB
最終ジャッジ日時 2023-10-21 19:08:50
合計ジャッジ時間 3,616 ms
ジャッジサーバーID
（参考情報） 		judge11 / judge12
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 1 ms
4,348 KB
testcase_01 AC 1 ms
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testcase_02 AC 1 ms
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testcase_03 AC 1 ms
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testcase_04 AC 1 ms
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testcase_05 AC 1 ms
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testcase_06 AC 1 ms
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testcase_07 AC 1 ms
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testcase_08 AC 1 ms
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testcase_09 AC 1 ms
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testcase_10 AC 1 ms
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testcase_11 AC 1 ms
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testcase_12 AC 1 ms
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testcase_13 AC 1 ms
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testcase_14 AC 1 ms
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testcase_15 AC 1 ms
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testcase_16 AC 1 ms
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testcase_17 AC 1 ms
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testcase_18 AC 1 ms
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testcase_19 AC 1 ms
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testcase_20 AC 1 ms
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testcase_21 AC 14 ms
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testcase_22 AC 12 ms
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testcase_23 AC 15 ms
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testcase_24 AC 7 ms
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testcase_25 AC 14 ms
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testcase_26 AC 10 ms
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testcase_27 AC 5 ms
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testcase_28 AC 3 ms
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testcase_29 AC 16 ms
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testcase_30 AC 16 ms
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testcase_31 AC 5 ms
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testcase_32 AC 14 ms
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testcase_33 AC 10 ms
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testcase_34 AC 15 ms
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testcase_35 AC 13 ms
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testcase_36 AC 16 ms
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testcase_37 AC 17 ms
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testcase_38 AC 17 ms
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ソースコード

diff # 

// 包除することを考えると
// A, B は交差しない
// A, B は非空
// A, B の要素はそれぞれx, y で割り切れる
// というのが計算できるといい
// f(x, y) が求めたいそれとしよう
// z = lcm(x, y) と置くと
// 3: N/z
// 1: N/x-N/z
// 2: N/y-N/z
//
// f(x, y) = 2^(C_x - C_z) * 2^(C_y - C_z) * 3^(C_z) - 2^(C_x) - 2^(C_y) + 1
// 後ろ3項の計算は容易
// 最初の項？？
// 2^(C_x) * 2^(C_y) * (3/4)^(C_lcm(x, y))
// C_x = floor(N / x)
// lcm(x, y) が厄介
// N/xの種類数は少ないけどもこれ活かすのかな
// いやlcm畳み込みで大体わかる
//

fn run() {
    input!(n: usize);
    let mut mo = vec![M::one(); n + 1];
    mo[0] = M::zero();
    enumerate_prime(n, |p| {
        for i in 1..=(n / p) {
            mo[i * p] = -mo[i * p];
        }
        for i in 1..=(n / (p * p)) {
            mo[i * p * p] = M::zero();
        }
    });
    let two = M::new(2);
    let p = M::new(3) * M::new(4).inv();
    let mut dp = vec![M::zero(); n + 1];
    for i in 1..=n {
        dp[i] = mo[i] * two.pow((n / i) as u64);
    }
    let mo_sum = mo.iter().fold(M::zero(), |s, a| s + *a);
    let sum = dp.iter().fold(M::zero(), |s, a| s + *a);
    enumerate_prime(n, |p| {
        for i in 1..=(n / p) {
            dp[i * p] = dp[i * p] + dp[i];
        }
    });
    dp.iter_mut().for_each(|dp| *dp = *dp * *dp);
    enumerate_prime(n, |p| {
        for i in (1..=(n / p)).rev() {
            dp[i * p] = dp[i * p] - dp[i];
        }
    });
    let mut ans = M::zero();
    ans += sum * sum - dp.iter().fold(M::zero(), |s, a| s + *a);
    ans += mo_sum * mo_sum;
    for i in 1..=n {
        ans += dp[i] * p.pow((n / i) as u64);
        ans -= mo[i] * two.pow((n / i) as u64) * mo_sum * M::new(2);
    }
    println!("{}", ans);
}

fn main() {
    run();
}

// ---------- begin enumerate prime ----------
fn enumerate_prime<F>(n: usize, mut f: F)
where
    F: FnMut(usize),
{
    assert!(1 <= n && n <= 5 * 10usize.pow(8));
    let batch = (n as f64).sqrt().ceil() as usize;
    let mut is_prime = vec![true; batch + 1];
    for i in (2..).take_while(|p| p * p <= batch) {
        if is_prime[i] {
            let mut j = i * i;
            while let Some(p) = is_prime.get_mut(j) {
                *p = false;
                j += i;
            }
        }
    }
    let mut prime = vec![];
    for (i, p) in is_prime.iter().enumerate().skip(2) {
        if *p && i <= n {
            f(i);
            prime.push(i);
        }
    }
    let mut l = batch + 1;
    while l <= n {
        let r = std::cmp::min(l + batch, n + 1);
        is_prime.clear();
        is_prime.resize(r - l, true);
        for &p in prime.iter() {
            let mut j = (l + p - 1) / p * p - l;
            while let Some(is_prime) = is_prime.get_mut(j) {
                *is_prime = false;
                j += p;
            }
        }
        for (i, _) in is_prime.iter().enumerate().filter(|p| *p.1) {
            f(i + l);
        }
        l += batch;
    }
}
// ---------- end enumerate prime ----------
// ---------- begin input macro ----------
// reference: https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
#[macro_export]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

#[macro_export]
macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};
    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

#[macro_export]
macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };
    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };
    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };
    ($iter:expr, bytes) => {
        read_value!($iter, String).bytes().collect::<Vec<u8>>()
    };
    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };
    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}
// ---------- end input macro ----------
// ---------- begin modint ----------
use std::marker::*;
use std::ops::*;

pub trait Modulo {
    fn modulo() -> u32;
}

pub struct ConstantModulo<const M: u32>;

impl<const M: u32> Modulo for ConstantModulo<{ M }> {
    fn modulo() -> u32 {
        M
    }
}

pub struct ModInt<T>(u32, PhantomData<T>);

impl<T> Clone for ModInt<T> {
    fn clone(&self) -> Self {
        Self::new_unchecked(self.0)
    }
}

impl<T> Copy for ModInt<T> {}

impl<T: Modulo> Add for ModInt<T> {
    type Output = ModInt<T>;
    fn add(self, rhs: Self) -> Self::Output {
        let mut v = self.0 + rhs.0;
        if v >= T::modulo() {
            v -= T::modulo();
        }
        Self::new_unchecked(v)
    }
}

impl<T: Modulo> AddAssign for ModInt<T> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<T: Modulo> Sub for ModInt<T> {
    type Output = ModInt<T>;
    fn sub(self, rhs: Self) -> Self::Output {
        let mut v = self.0 - rhs.0;
        if self.0 < rhs.0 {
            v += T::modulo();
        }
        Self::new_unchecked(v)
    }
}

impl<T: Modulo> SubAssign for ModInt<T> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<T: Modulo> Mul for ModInt<T> {
    type Output = ModInt<T>;
    fn mul(self, rhs: Self) -> Self::Output {
        let v = self.0 as u64 * rhs.0 as u64 % T::modulo() as u64;
        Self::new_unchecked(v as u32)
    }
}

impl<T: Modulo> MulAssign for ModInt<T> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<T: Modulo> Neg for ModInt<T> {
    type Output = ModInt<T>;
    fn neg(self) -> Self::Output {
        if self.is_zero() {
            Self::zero()
        } else {
            Self::new_unchecked(T::modulo() - self.0)
        }
    }
}

impl<T> std::fmt::Display for ModInt<T> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T> std::fmt::Debug for ModInt<T> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T> Default for ModInt<T> {
    fn default() -> Self {
        Self::zero()
    }
}

impl<T: Modulo> std::str::FromStr for ModInt<T> {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

impl<T: Modulo> From<usize> for ModInt<T> {
    fn from(val: usize) -> ModInt<T> {
        ModInt::new_unchecked((val % T::modulo() as usize) as u32)
    }
}

impl<T: Modulo> From<u64> for ModInt<T> {
    fn from(val: u64) -> ModInt<T> {
        ModInt::new_unchecked((val % T::modulo() as u64) as u32)
    }
}

impl<T: Modulo> From<i64> for ModInt<T> {
    fn from(val: i64) -> ModInt<T> {
        let mut v = ((val % T::modulo() as i64) + T::modulo() as i64) as u32;
        if v >= T::modulo() {
            v -= T::modulo();
        }
        ModInt::new_unchecked(v)
    }
}

impl<T> ModInt<T> {
    pub fn new_unchecked(n: u32) -> Self {
        ModInt(n, PhantomData)
    }
    pub fn zero() -> Self {
        ModInt::new_unchecked(0)
    }
    pub fn one() -> Self {
        ModInt::new_unchecked(1)
    }
    pub fn is_zero(&self) -> bool {
        self.0 == 0
    }
}

impl<T: Modulo> ModInt<T> {
    pub fn new(d: u32) -> Self {
        ModInt::new_unchecked(d % T::modulo())
    }
    pub fn pow(&self, mut n: u64) -> Self {
        let mut t = Self::one();
        let mut s = *self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(&self) -> Self {
        assert!(!self.is_zero());
        self.pow(T::modulo() as u64 - 2)
    }
    pub fn fact(n: usize) -> Self {
        (1..=n).fold(Self::one(), |s, a| s * Self::from(a))
    }
    pub fn perm(n: usize, k: usize) -> Self {
        if k > n {
            return Self::zero();
        }
        ((n - k + 1)..=n).fold(Self::one(), |s, a| s * Self::from(a))
    }
    pub fn binom(n: usize, k: usize) -> Self {
        if k > n {
            return Self::zero();
        }
        let k = k.min(n - k);
        let mut nu = Self::one();
        let mut de = Self::one();
        for i in 0..k {
            nu *= Self::from(n - i);
            de *= Self::from(i + 1);
        }
        nu * de.inv()
    }
}
// ---------- end modint ----------
// ---------- begin precalc ----------
pub struct Precalc<T> {
    fact: Vec<ModInt<T>>,
    ifact: Vec<ModInt<T>>,
    inv: Vec<ModInt<T>>,
}

impl<T: Modulo> Precalc<T> {
    pub fn new(n: usize) -> Precalc<T> {
        let mut inv = vec![ModInt::one(); n + 1];
        let mut fact = vec![ModInt::one(); n + 1];
        let mut ifact = vec![ModInt::one(); n + 1];
        for i in 2..=n {
            fact[i] = fact[i - 1] * ModInt::new_unchecked(i as u32);
        }
        ifact[n] = fact[n].inv();
        if n > 0 {
            inv[n] = ifact[n] * fact[n - 1];
        }
        for i in (1..n).rev() {
            ifact[i] = ifact[i + 1] * ModInt::new_unchecked((i + 1) as u32);
            inv[i] = ifact[i] * fact[i - 1];
        }
        Precalc { fact, ifact, inv }
    }
    pub fn inv(&self, n: usize) -> ModInt<T> {
        assert!(n > 0);
        self.inv[n]
    }
    pub fn fact(&self, n: usize) -> ModInt<T> {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModInt<T> {
        self.ifact[n]
    }
    pub fn perm(&self, n: usize, k: usize) -> ModInt<T> {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[n - k]
    }
    pub fn binom(&self, n: usize, k: usize) -> ModInt<T> {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end precalc ----------

type M = ModInt<ConstantModulo<998_244_353>>;
0