Problem:
Score : 600 points

Problem Statement

Nagase is a top student in high school. One day, she's analyzing some properties of special sets of positive integers.

She thinks that a set S = \{a_{1}, a_{2}, ..., a_{N}\} of distinct positive integers is called special if for all 1 \leq i \leq N, the gcd (greatest common divisor) of a_{i} and the sum of the remaining elements of S is not 1.

Nagase wants to find a special set of size N. However, this task is too easy, so she decided to ramp up the difficulty. Nagase challenges you to find a special set of size N such that the gcd of all elements are 1 and the elements of the set does not exceed 30000.

Constraints

3 \leq N \leq 20000

Input

Input is given from Standard Input in the following format:

N

Output

Output N space-separated integers, denoting the elements of the set S. S must satisfy the following conditions :

The elements must be distinct positive integers not exceeding 30000.

The gcd of all elements of S is 1, i.e. there does not exist an integer d > 1 that divides all elements of S.

S is a special set.

If there are multiple solutions, you may output any of them. The elements of S may be printed in any order. It is guaranteed that at least one solution exist under the given contraints.

Sample Input 1

3

Sample Output 1

2 5 63

\{2, 5, 63\} is special because gcd(2, 5 + 63) = 2, gcd(5, 2 + 63) = 5, gcd(63, 2 + 5) = 7. Also, gcd(2, 5, 63) = 1. Thus, this set satisfies all the criteria.

Note that \{2, 4, 6\} is not a valid solution because gcd(2, 4, 6) = 2 > 1.

Sample Input 2

4

Sample Output 2

2 5 20 63

\{2, 5, 20, 63\} is special because gcd(2, 5 + 20 + 63) = 2, gcd(5, 2 + 20 + 63) = 5, gcd(20, 2 + 5 + 63) = 10, gcd(63, 2 + 5 + 20) = 9. Also, gcd(2, 5, 20, 63) = 1. Thus, this set satisfies all the criteria.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_macros)]
macro_rules! parse_line {
    ( $t:ty ) => (
    {
        let mut line = String::new();
        ::std::io::stdin().read_line(&mut line).unwrap();
        let mut iter = line.split_whitespace();
        iter.next().unwrap().parse::<$t>().unwrap()
    }
);

( $( $t:ty), +) => (
{
    let mut line = String::new();
    ::std::io::stdin().read_line(&mut line).unwrap();
    let mut iter = line.split_whitespace();
    ( $(iter.next().unwrap().parse::<$t>().unwrap()),* )
}
    );
}

#[allow(unused_macros)]
macro_rules! read_line {
    () => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            line.pop();
            line
        }
    )
}

#[allow(unused_macros)]
macro_rules! parse_vec {
    ( $t:ty ) => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            let iter = line.split_whitespace();
            iter.map(|v| v.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    )
}

fn solve(n: u32) -> Vec<u32> {

    if n == 3 {
        return vec![2,5,63];
    } else if n == 4 {
        return vec![2,5,20,63];
    } else if n == 5 {
        return vec![2,5,20,30,63];
    }

    let mut k = 0;
    let mut v = Vec::new();
    let mut sum = 0;
    for i in 1..n+1 {
        let x = match i % 4 {
            1 => { 6*k + 2},
            2 => { 6*k + 3},
            3 => { 6*k + 4},
            0 => {
                k += 1;
                6*k
            },
            _ => unreachable!()
        };
        v.push(x);
        sum += x;
    }

    match sum % 6 {
        0 => {},
        2 => {
            v[4] = 30000;
        },
        5 => {
            v[5] = 29998;
        },
        3 => {
            v[5] = 30000;
        },
        _ => unreachable!()
    }
    v
}

fn main() {
    let n = parse_line!(u32);
    println!("{}", solve(n).iter().map(|i| i.to_string()).collect::<Vec<String>>().join(" "));
}

#[cfg(test)]
mod unit_test {
    use super::*;

    fn gcd(a: u64, b: u64) -> u64 {
        let x = a % b;
        if x == 0 {
            b
        } else {
            gcd(b, x)
        }
    }

    #[test]
    fn gcd_test() {
        assert_eq!(gcd(2,3), 1);
        assert_eq!(gcd(63, 49) , 7);
    }

    fn check(v: Vec<u32>) {
        let sum = v.iter().fold(0 as u64, |sum, x| sum + *x as u64);
        for x in v.iter() {
            assert_ne!(gcd(*x as u64, sum),1);
        }
        assert_eq!(v.iter().fold(v[0] as u64, |g, x| gcd(g as u64, *x as u64)), 1);

        let m = v.iter().fold(0, |max, x| if max < *x { *x} else { max });
        println!("m = {}", m);
        assert!(m <= 30000);
    }

    #[test]
    fn solve_test() {
        for n in 3..100 {
            check(solve(n));
        }

        check(solve(20000));
        check(solve(19999));
        check(solve(19998));
    }
}