Prime factorization base on sieve of Eratosthenes.

primeFactorizationBaseOnEratosthenes.js

function primeFactorizationBaseOnEratosthenes(x) {
    if (x === 0)
        return;

    let seq = Array(x + 1); //index:0~x
    let sqrt = Math.sqrt(x);

    seq[1] = 1;

    //Base on sieve of Eratosthene.Get store smallest prime factor.
    //if index is prime,the value is empty.
    for (let i = 2; i <= sqrt; i++) {
        if (!seq[i]) {
            for (let j = Math.pow(i, 2); j <= x; j += i) {
                seq[j] = i; //store smallest prime factor(index is composite).
            }
        }
    }

    //prime factorization
    let f = [];
    while (x != 1) {
        let a = seq[x] ? seq[x] : x; //if value is empty,assign x. 
        f.push(a);
        x /= a;
    }
    return f;
}

for (let i = 0; i <= 20; i++) {
    console.log(i, primeFactorizationBaseOnEratosthenes(i));
}

/*
0 undefined
1 []
2 [2]
3 [3]
4 [2, 2]
5 [5]
6 [2, 3]
7 [7]
8 [2, 2, 2]
9 [3, 3]
10 [2, 5]
11 [11]
12 [3, 2, 2]
13 [13]
14 [2, 7]
15 [3, 5]
16 [2, 2, 2, 2]
17 [17]
18 [3, 2, 3]
19 [19]
20 [2, 2, 5]
*/

Python implementation: https://gist.github.com/ShenTengTu/bc1cba365e46971c6f62aba8fc803cfa