// 商を計算したい感 // 1 <= i <= N, 1 <= j <= M // i/j >= q // なる数を計算したい? // // N について商列挙すると? // l, r, q // l <= d <= r で floor(n/d) = q // // sum_{l <= j <= r} sum_{1 <= i <= N} floor(i/j) * j // i < qj について // sum_j sum_{1 <= i < qj} .. // = sum_j (q-1)q/2 * j // qj <= i について // sum_j sum_i qj // sum_j (N - qj + 1) * q * j use std::io::Write; use std::collections::*; type Map = BTreeMap; type Set = BTreeSet; type Deque = VecDeque; fn main() { input! { n: u64, m: u64, } let f = |n: u64| -> M { M::from(n) * M::from(n + 1) * M::new(2).inv() }; let g = |n: u64| -> M { M::from(n) * M::from(n + 1) * M::from(2 * n + 1) * M::new(6).inv() }; let mut ans = f(n) * M::from(m); quot_range_unorderd(n, |l, r, q| { if m < l { return; } let r = r.min(m); ans -= f(q - 1) * (g(r) - g(l - 1)); ans -= M::from(n + 1) * M::from(q) * (f(r) - f(l - 1)); ans += M::from(q) * M::from(q) * (g(r) - g(l - 1)); }); println!("{}", ans); } // ---------- 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, $t)).collect::>() }; ($iter:expr, chars) => { read_value!($iter, String).chars().collect::>() }; ($iter:expr, bytes) => { read_value!($iter, String).bytes().collect::>() }; ($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; impl Modulo for ConstantModulo<{ M }> { fn modulo() -> u32 { M } } pub struct ModInt(u32, PhantomData); impl Clone for ModInt { fn clone(&self) -> Self { Self::new_unchecked(self.0) } } impl Copy for ModInt {} impl Add for ModInt { type Output = ModInt; 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 AddAssign for ModInt { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs; } } impl Sub for ModInt { type Output = ModInt; 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 SubAssign for ModInt { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs; } } impl Mul for ModInt { type Output = ModInt; 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 MulAssign for ModInt { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs; } } impl Neg for ModInt { type Output = ModInt; fn neg(self) -> Self::Output { if self.is_zero() { Self::zero() } else { Self::new_unchecked(T::modulo() - self.0) } } } impl std::fmt::Display for ModInt { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.0) } } impl std::fmt::Debug for ModInt { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.0) } } impl Default for ModInt { fn default() -> Self { Self::zero() } } impl std::str::FromStr for ModInt { type Err = std::num::ParseIntError; fn from_str(s: &str) -> Result { let val = s.parse::()?; Ok(ModInt::new(val)) } } impl From for ModInt { fn from(val: usize) -> ModInt { ModInt::new_unchecked((val % T::modulo() as usize) as u32) } } impl From for ModInt { fn from(val: u64) -> ModInt { ModInt::new_unchecked((val % T::modulo() as u64) as u32) } } impl From for ModInt { fn from(val: i64) -> ModInt { 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 ModInt { 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 ModInt { 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 { fact: Vec>, ifact: Vec>, inv: Vec>, } impl Precalc { pub fn new(n: usize) -> Precalc { 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 { assert!(n > 0); self.inv[n] } pub fn fact(&self, n: usize) -> ModInt { self.fact[n] } pub fn ifact(&self, n: usize) -> ModInt { self.ifact[n] } pub fn perm(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[n - k] } pub fn binom(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[k] * self.ifact[n - k] } } // ---------- end precalc ---------- type M = ModInt>; // ---------- begin floor sum ---------- // sum_{i = 0}^{n - 1} floor((ai + b) / m) pub fn floor_sum(n: u64, m: u64, mut a: u64, mut b: u64) -> u64 { assert!(n <= 10u64.pow(9)); assert!(1 <= m && m <= 10u64.pow(9)); let mut ans = 0; ans += a / m * n * (n - 1) / 2 + b / m * n; a %= m; b %= m; let p = a * n + b; if p >= m { ans += floor_sum(p / m, a, m, p % m); } ans } // ---------- end floor sum ---------- // ---------- begin quot_range ---------- // 商列挙 // n を与えると [1, n] を (l, r, q) な組に分解する // (l, r, q): x \in [l, r] <=> floor(n / x) = q // // 実装が3通りある // quot_range_orderd // (l, r, q) を (1, 1, n), (2, 2, n / 2)... の順に返す // 除算 ceil(sqrt(n)) 回 // メモリ O(sqrt(n)) // // quot_range_unorderd // (l, r, q) を (1, 1, n), (ceil(n/2), n, 1), ...の順に返す // 除算 ceil(sqrt(n)) 回 // メモリO(1) // // quot_range_nanka // (l, r, q) を (1, 1, n), (2, 2, n / 2)... の順に返す // 除算 2 * ceil(sqrt(n)) 回 // メモリ O(1) // // どれがいいかはよくわからない // 順序不定でいいならunorderd // 除算が極めて重いならorderd // ML が気になるならnanka // // todo: u32 だけじゃなくu64 でも #[allow(dead_code)] pub fn quot_range_orderd(n: u32, mut f: F) where F: FnMut(u32, u32, u32), { let mut p = Vec::with_capacity((n as f64).sqrt().ceil() as usize + 1); for d in 1.. { let q = n / d; p.push((d, q)); if q < d { break; } f(d, d, q); if q == d { break; } } for p in p.windows(2).rev() { let (l, r, q) = (p[1].1 + 1, p[0].1, p[0].0); f(l, r, q); } } #[allow(dead_code)] pub fn quot_range_unorderd(n: u64, mut f: F) where F: FnMut(u64, u64, u64) { let mut pre = (0, 0); for d in 1.. { let q = n / d; if pre != (0, 0) { let (q, l, r) = (pre.0, q + 1, pre.1); f(l, r, q); } if q < d { break; } pre = (d, q); f(d, d, q); if q == d { break; } } } #[allow(dead_code)] pub fn quot_range_nanka(n: u32, mut f: F) where F: FnMut(u32, u32, u32) { let mut l = 1; while l * l <= n { let q = n / l; f(l, l, q); l += 1; } let mut q = n / l; while q > 0 { let r = n / q; f(l, r, q); l = r + 1; q -= 1; } } // ---------- begin quot_range ----------