Problem:
Score : 1200 points

Problem Statement

There are infinitely many cards, numbered 1, 2, 3, ...
Initially, Cards x_1, x_2, ..., x_N are face up, and the others are face down.

Snuke can perform the following operation repeatedly:

Select a prime p greater than or equal to 3. Then, select p consecutive cards and flip all of them.

Snuke's objective is to have all the cards face down.
Find the minimum number of operations required to achieve the objective.

Constraints

1 ≤ N ≤ 100

1 ≤ x_1 < x_2 < ... < x_N ≤ 10^7

Input

Input is given from Standard Input in the following format:

N
x_1 x_2 ... x_N

Output

Print the minimum number of operations required to achieve the objective.

Sample Input 1

2
4 5

Sample Output 1

2

Below is one way to achieve the objective in two operations:

Select p = 5 and flip Cards 1, 2, 3, 4 and 5.

Select p = 3 and flip Cards 1, 2 and 3.

Sample Input 2

9
1 2 3 4 5 6 7 8 9

Sample Output 2

3

Below is one way to achieve the objective in three operations:

Select p = 3 and flip Cards 1, 2 and 3.

Select p = 3 and flip Cards 4, 5 and 6.

Select p = 3 and flip Cards 7, 8 and 9.

Sample Input 3

2
1 10000000

Sample Output 3

4

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

#![allow(unused_imports, unused_variables, dead_code)]
use std::io::*;
use std::fmt::*;
use std::str::*;
use std::cmp::*;
use std::collections::*;

pub trait InputValue {
    fn parse(s: &str) -> Self;
}

pub fn read<T: InputValue>() -> T {
    let mut buf = String::new();
    let _ = stdin().read_line(&mut buf);
    T::parse(&buf.trim())
}

pub fn readnc<T: InputValue>() -> Vec<T> {
    let mut vec = vec![];
    let line: String = read();
    for token in line.split_whitespace() {
        vec.push(T::parse(token));
    }
    vec
}

pub fn readn<T: InputValue>(n: usize) -> Vec<T> {
    let mut vec = vec![];
    for _ in 0..n {
        vec.push(read());
    }
    vec
}

macro_rules! parse_single_value {
    ($($t:ty),*) => {
        $(
            impl InputValue for $t {
                fn parse(s: &str) -> $t { s.parse().unwrap() }
            }
        )*
	}
}
parse_single_value!(i32, i64, f32, f64, usize, String);

macro_rules! parse_tuple {
	($($t:ident),*) => {
		impl<$($t),*> InputValue for ($($t),*) where $($t: InputValue),* {
			fn parse(s: &str) -> ($($t),*) {
				let mut tokens = s.split_whitespace();
				let t = ($($t::parse(tokens.next().unwrap())),*);
				t
			}
		}
	}
}
parse_tuple!(A, B);
parse_tuple!(A, B, C);

// ===
const MAX_FLOW: i32 = 100000000;

#[derive(Clone, Copy)]
struct Edge {
    from: usize,
    to: usize,
    cap: i32,
    maxcap: i32
}

impl Edge {
    fn goto(&self, from: usize) -> usize {
        self.from + self.to - from
    }

    fn cap(&self, from: usize) -> i32 {
        if self.from == from {
            self.cap
        } else {
            self.maxcap - self.cap
        }
    }
}

struct FlowGraph {
    n: usize,
    graph: Vec<Vec<usize>>,
    edges: Vec<Edge>,
    level: Vec<i32>,
    itr: Vec<usize>
}

impl FlowGraph {
    fn new(n: usize) -> Self {
        let graph = vec![vec![]; n];
        let level = vec![0; n];
        let itr = vec![0; n];
        FlowGraph { n: n, graph: graph, edges: vec![], level: level, itr: itr }
    }

    fn edge(&mut self, from: usize, to: usize, cap: i32) {
        let e = Edge { from: from, to: to, cap: cap, maxcap: cap };
        let l = self.edges.len();
        self.graph[from].push(l);
        self.graph[to].push(l);
        self.edges.push(e);
    }

    fn dfs(&mut self, now: usize, to: usize, flow: i32) -> i32 {
        if now == to {
            return flow;
        }
        for i in self.itr[now]..self.graph[now].len() {
            self.itr[now] = i;
            let eidx = self.graph[now][i];

            let e = self.edges[eidx];
            let next = e.goto(now);
            let ecap = e.cap(now);
            if ecap > 0 && self.level[now] < self.level[next] {
                let d = self.dfs(next, to, min(flow, ecap));
                if d >= 1 {
                    let e = &mut self.edges[eidx];
                    if e.from == now {
                        e.cap -= d;
                    } else {
                        e.cap += d;
                    }
                    return d;
                }
            }

        }
        return 0;
    }

    fn bfs(&mut self, from: usize) {
        for i in 0..self.n {
            self.level[i] = -1;
        }
        self.level[from] = 0;
        let mut q: VecDeque<usize> = VecDeque::new();
        q.push_back(from);
        while let Some(idx) = q.pop_front() {
            for &eidx in &self.graph[idx] {
                let e = &self.edges[eidx];
                let to = e.goto(idx);
                if e.cap(idx) >= 1 && self.level[to] == -1 {
                    self.level[to] = self.level[idx] + 1;
                    q.push_back(to);
                }
            }
        }
    }

    fn max_flow(&mut self, from: usize, to: usize) -> i32 {
        let mut flow = 0;
        loop {
            self.bfs(from);
            if self.level[to] < 0 {
                return flow;
            }
            for i in 0..self.n {
                self.itr[i] = 0;
            }
            loop {
                let f = self.dfs(from, to, MAX_FLOW);
                if f <= 0 {
                    break;
                }
                flow += f;
            }
        }
    }
}

// ===

fn testflow() {
    let mut flow = FlowGraph::new(5);
    flow.edge(0, 1, 1);
    flow.edge(1, 2, 1);
    flow.edge(3, 0, 1);
    flow.edge(2, 4, 1);
    flow.edge(3, 4, 1);
    println!("f={}", flow.max_flow(0, 4));
}

fn prime(n: usize) -> Vec<bool> {
    let mut ret = vec![true; n];
    ret[0] = false;
    ret[1] = false;
    for i in 2..n {
        if !ret[i] {
            continue
        }
        let mut ii = i * i;
        while ii < n {
            ret[ii] = false;
            ii += i;
        }
    }
    ret
}

fn main() {
    let n: usize = read();
    let x: Vec<i32> = readnc();

    let mut fpos = vec![];
    fpos.push(x[0]);

    for i in 1..n {
        if x[i-1] + 1 < x[i] {
            fpos.push(x[i-1]+1);
            fpos.push(x[i])
        }
    }
    fpos.push(x[n-1]+1);

    let isp = prime(10000100);

    let n = fpos.len();
    let mut even = 0;
    let mut odd = 0;
    for &f in &fpos {
        if f % 2 == 0 {
            even += 1;
        } else {
            odd += 1;
        }
    }

    let mut flow = FlowGraph::new(n + 2);
    let source = n;
    let sink = n + 1;
    for i in 0..n {
        for j in 0..n {
            let d = (fpos[i] - fpos[j]).abs() as usize;
            if i != j && fpos[i] % 2 == 0 && fpos[j] % 2 == 1 && isp[d] {
                flow.edge(i, j, 1);
            }
        }
    }
    for i in 0..n {
        if fpos[i] % 2 == 0 {
            flow.edge(source, i, 1);
        } else {
            flow.edge(i, sink, 1);
        }
    }

    let pair = flow.max_flow(source, sink);
    let even_left = even - pair;
    let odd_left = odd - pair;
    println!("{}", pair + (even_left / 2) * 2 + (odd_left / 2) * 2 + max(0, odd_left % 2) * 3);
}