Problem:
Score : 500 points

Problem Statement

Given are N positive integers A_1,...,A_N.

Consider positive integers B_1, ..., B_N that satisfy the following condition.

Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds.

Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N.

Since the answer can be enormous, print the sum modulo (10^9 +7).

Constraints

1 \leq N \leq 10^4

1 \leq A_i \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7).

Sample Input 1

3
2 3 4

Sample Output 1

13

Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied.

Sample Input 2

5
12 12 12 12 12

Sample Output 2

5

We can let all B_i be 1.

Sample Input 3

3
1000000 999999 999998

Sample Output 3

996989508

Print the sum modulo (10^9+7).

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let n = read::<usize>();
    let a = read_vec::<i64>();
    let a_max = *a.iter().max().unwrap() + 1;
    let mut primes = vec![];
    for i in 1..a_max as i64 {
        if i * i > a_max {
            primes = get_primes(i + 10);
            break;
        }
    }
    let mut dict_all = HashMap::new();
    for i in 0..n {
        let dict = factorize(a[i], &primes);
        for (k, v) in dict {
            if dict_all.contains_key(&k) {
                if v > dict_all[&k] {
                    *dict_all.get_mut(&k).unwrap() = v;
                }
            } else {
                dict_all.insert(k, v);
            }
        }
    }
    debug!(dict_all);

    let mut ans = Modulo(0);
    for i in 0..n {
        let dict = factorize(a[i], &primes);
        let mut temp = Modulo(1);
        for (k, v) in dict_all.iter() {
            if dict.contains_key(k) {
                temp *= Modulo(*k).pow(*v - dict[k]);
            } else {
                temp *= Modulo(*k).pow(*v);
            }
        }
        ans += temp;
    }
    println!("{}", ans);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

fn get_primes(n: i64) -> Vec<i64> {
    let mut is_prime = vec![true; n as usize + 1];
    let mut primes = Vec::new();
    is_prime[0] = false;
    is_prime[1] = false;

    for i in 2..n + 1 {
        if is_prime[i as usize] {
            primes.push(i);
            let mut j = 2 * i;
            while j < n {
                is_prime[j as usize] = false;
                j += i;
            }
        }
    }
    primes
}

fn factorize(mut num: i64, primes: &Vec<i64>) -> std::collections::HashMap<i64, i64> {
    // max_primes >= (num)^(1/2)
    let mut dict = std::collections::HashMap::new();
    for &p in primes.iter() {
        while num % p == 0 {
            *dict.entry(p).or_insert(0) += 1;
            num /= p;
        }
        if num == 1 {
            break;
        }
    }
    if num != 1 {
        dict.insert(num, 1);
    }
    dict
}

#[derive(Clone, Copy, Debug, Default, Eq, Hash, Ord, PartialEq, PartialOrd)]
struct Modulo(i64);
static mut MODULUS: i64 = 1000000007;
impl Modulo {
    fn set_modulus(m: i64) {
        unsafe {
            MODULUS = m;
        }
    }
    fn get_modulus() -> i64 {
        unsafe { MODULUS }
    }
    fn new(x: i64) -> Modulo {
        let m = Modulo::get_modulus();
        if x < 0 {
            Modulo(x % m + m)
        } else if x < m {
            Modulo(x)
        } else {
            Modulo(x % m)
        }
    }
    fn pow(self, p: i64) -> Modulo {
        if p == 0 {
            Modulo(1)
        } else {
            let mut t = self.pow(p / 2);
            t *= t;
            if p & 1 == 1 {
                t *= self;
            }
            t
        }
    }
    fn inv(self) -> Modulo {
        self.pow(Modulo::get_modulus() - 2)
    }
}
impl std::fmt::Display for Modulo {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        self.0.fmt(f)
    }
}
impl std::ops::AddAssign for Modulo {
    fn add_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
impl std::ops::MulAssign for Modulo {
    fn mul_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 *= other.0;
        self.0 %= m;
    }
}
impl std::ops::SubAssign for Modulo {
    fn sub_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += m - other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
macro_rules! impl_modulo_ops {
    ($imp:ident, $method:ident, $assign_imp:ident, $assign_method:ident) => {
        impl<'a> std::ops::$assign_imp<&'a Modulo> for Modulo {
            fn $assign_method(&mut self, other: &'a Modulo) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp for Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a> std::ops::$imp<&'a Modulo> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a Modulo) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b Modulo) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
        impl std::ops::$assign_imp<i64> for Modulo {
            fn $assign_method(&mut self, other: i64) {
                std::ops::$assign_imp::$assign_method(self, Modulo::new(other));
            }
        }
        impl<'a> std::ops::$assign_imp<&'a i64> for Modulo {
            fn $assign_method(&mut self, other: &'a i64) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp<i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<&'a i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a i64) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a> std::ops::$imp<i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b i64) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
    };
}
impl_modulo_ops!(Add, add, AddAssign, add_assign);
impl_modulo_ops!(Mul, mul, MulAssign, mul_assign);
impl_modulo_ops!(Sub, sub, SubAssign, sub_assign);