# Prime Factorization using Sieve O(log n) for multiple queries

Last Updated : 04 May, 2023

We can calculate the prime factorization of a number “n” in O(sqrt(n)) as discussed here. But O(sqrt n) method times out when we need to answer multiple queries regarding prime factorization.
In this article, we study an efficient method to calculate the prime factorization using O(n) space and O(log n) time complexity with pre-computation allowed.

Prerequisites : Sieve of Eratosthenes, Least prime factor of numbers till n.

Recommended Practice

Approach:

The main idea is to precompute the Smallest Prime Factor (SPF) for each number from 1 to MAXN using the sieve function. SPF is the smallest prime number that divides a given number without leaving a remainder. Then, the getFactorization function uses the precomputed SPF array to find the prime factorization of the given number by repeatedly dividing the number by its SPF until it becomes 1.

To calculate to smallest prime factor for every number we will use the sieve of eratosthenes. In the original Sieve, every time we mark a number as not prime, we store the corresponding smallest prime factor for that number (Refer this article for better understanding).

Now, after we are done with precalculating the smallest prime factor for every number we will divide our number n (whose prime factorization is to be calculated) by its corresponding smallest prime factor till n becomes 1.

```Pseudo Code for prime factorization assuming
SPFs are computed :

PrimeFactors[] // To store result

i = 0  // Index in PrimeFactors

while n != 1 :

// SPF : smallest prime factor
PrimeFactors[i] = SPF[n]
i++
n = n / SPF[n]```

Step-by-step approach of above idea:

• Defines a constant MAXN equal to 100001.
• An integer array spf of size MAXN is declared. This array will store the smallest prime factor for each number up to MAXN.
• A function sieve() is defined to calculate the smallest prime factor of every number up to MAXN using the Sieve of Eratosthenes algorithm.
• The smallest prime factor for the number 1 is set to 1.
• The smallest prime factor for every number from 2 to MAXN is initialized to be the number itself.
• The smallest prime factor for every even number from 4 to MAXN is set to 2.
• The smallest prime factor for every odd number from 3 to the square root of MAXN is calculated by iterating over all odd numbers and checking if the number is prime.
• If a number i is prime, then the smallest prime factor for all numbers divisible by i is set to i.
• A function getFactorization(int x) is defined to return the prime factorization of a given integer x using the spf array.
• The getFactorization(int x) function finds the smallest prime factor of x, pushes it to a vector, and updates x to be the quotient of x divided by its smallest prime factor. This process continues until x becomes 1, at which point the vector of prime factors is returned.
• In the main() function, the sieve() function is called to precalculate the smallest prime factor of every number up to MAXN. Then, the prime factorization of a sample integer x is found using the getFactorization(int x) function, and the result is printed to the console.

The implementation for the above method is given below :

## C++

 `// C++ program to find prime factorization of a` `// number n in O(Log n) time with precomputation` `// allowed.` `#include "bits/stdc++.h"` `using` `namespace` `std;`   `#define MAXN 100001`   `// stores smallest prime factor for every number` `int` `spf[MAXN];`   `// Calculating SPF (Smallest Prime Factor) for every` `// number till MAXN.` `// Time Complexity : O(nloglogn)` `void` `sieve()` `{` `    ``spf[1] = 1;` `    ``for` `(``int` `i = 2; i < MAXN; i++)`   `        ``// marking smallest prime factor for every` `        ``// number to be itself.` `        ``spf[i] = i;`   `    ``// separately marking spf for every even` `    ``// number as 2` `    ``for` `(``int` `i = 4; i < MAXN; i += 2)` `        ``spf[i] = 2;`   `    ``for` `(``int` `i = 3; i * i < MAXN; i++) {` `        ``// checking if i is prime` `        ``if` `(spf[i] == i) {` `            ``// marking SPF for all numbers divisible by i` `            ``for` `(``int` `j = i * i; j < MAXN; j += i)`   `                ``// marking spf[j] if it is not` `                ``// previously marked` `                ``if` `(spf[j] == j)` `                    ``spf[j] = i;` `        ``}` `    ``}` `}`   `// A O(log n) function returning primefactorization` `// by dividing by smallest prime factor at every step` `vector<``int``> getFactorization(``int` `x)` `{` `    ``vector<``int``> ret;` `    ``while` `(x != 1) {` `        ``ret.push_back(spf[x]);` `        ``x = x / spf[x];` `    ``}` `    ``return` `ret;` `}`   `// driver program for above function` `int` `main(``int` `argc, ``char` `const``* argv[])` `{` `    ``// precalculating Smallest Prime Factor` `    ``sieve();` `    ``int` `x = 12246;` `    ``cout << ``"prime factorization for "` `<< x << ``" : "``;`   `    ``// calling getFactorization function` `    ``vector<``int``> p = getFactorization(x);`   `    ``for` `(``int` `i = 0; i < p.size(); i++)` `        ``cout << p[i] << ``" "``;` `    ``cout << endl;` `    ``return` `0;` `}`

## Java

 `// Java program to find prime factorization of a` `// number n in O(Log n) time with precomputation` `// allowed.`   `import` `java.util.Vector;`   `class` `Test` `{` `    ``static` `final` `int` `MAXN = ``100001``;` `     `  `    ``// stores smallest prime factor for every number` `    ``static` `int` `spf[] = ``new` `int``[MAXN];` `     `  `    ``// Calculating SPF (Smallest Prime Factor) for every` `    ``// number till MAXN.` `    ``// Time Complexity : O(nloglogn)` `    ``static` `void` `sieve()` `    ``{` `        ``spf[``1``] = ``1``;` `        ``for` `(``int` `i=``2``; i getFactorization(``int` `x)` `    ``{` `        ``Vector ret = ``new` `Vector<>();` `        ``while` `(x != ``1``)` `        ``{` `            ``ret.add(spf[x]);` `            ``x = x / spf[x];` `        ``}` `        ``return` `ret;` `    ``}` `     `  `    ``// Driver method` `    ``public` `static` `void` `main(String args[])` `    ``{` `        ``// precalculating Smallest Prime Factor` `        ``sieve();` `        ``int` `x = ``12246``;` `        ``System.out.print(``"prime factorization for "` `+ x + ``" : "``);` `     `  `        ``// calling getFactorization function` `        ``Vector p = getFactorization(x);` `     `  `        ``for` `(``int` `i=``0``; i

## Python3

 `# Python3 program to find prime factorization ` `# of a number n in O(Log n) time with ` `# precomputation allowed.` `import` `math as mt`   `MAXN ``=` `100001`   `# stores smallest prime factor for` `# every number` `spf ``=` `[``0` `for` `i ``in` `range``(MAXN)]`   `# Calculating SPF (Smallest Prime Factor) ` `# for every number till MAXN.` `# Time Complexity : O(nloglogn)` `def` `sieve():` `    ``spf[``1``] ``=` `1` `    ``for` `i ``in` `range``(``2``, MAXN):` `        `  `        ``# marking smallest prime factor ` `        ``# for every number to be itself.` `        ``spf[i] ``=` `i`   `    ``# separately marking spf for ` `    ``# every even number as 2` `    ``for` `i ``in` `range``(``4``, MAXN, ``2``):` `        ``spf[i] ``=` `2`   `    ``for` `i ``in` `range``(``3``, mt.ceil(mt.sqrt(MAXN))):` `        `  `        ``# checking if i is prime` `        ``if` `(spf[i] ``=``=` `i):` `            `  `            ``# marking SPF for all numbers` `            ``# divisible by i` `            ``for` `j ``in` `range``(i ``*` `i, MAXN, i): ` `                `  `                ``# marking spf[j] if it is ` `                ``# not previously marked` `                ``if` `(spf[j] ``=``=` `j):` `                    ``spf[j] ``=` `i`   `# A O(log n) function returning prime ` `# factorization by dividing by smallest ` `# prime factor at every step` `def` `getFactorization(x):` `    ``ret ``=` `list``()` `    ``while` `(x !``=` `1``):` `        ``ret.append(spf[x])` `        ``x ``=` `x ``/``/` `spf[x]`   `    ``return` `ret`   `# Driver code`   `# precalculating Smallest Prime Factor` `sieve()` `x ``=` `12246` `print``(``"prime factorization for"``, x, ``": "``,` `                                ``end ``=` `"")`   `# calling getFactorization function` `p ``=` `getFactorization(x)`   `for` `i ``in` `range``(``len``(p)):` `    ``print``(p[i], end ``=` `" "``)`   `# This code is contributed ` `# by Mohit kumar 29`

## C#

 `// C# program to find prime factorization of a` `// number n in O(Log n) time with precomputation` `// allowed.` `using` `System;` `using` `System.Collections;`   `class` `GFG` `{` `    ``static` `int` `MAXN = 100001;` `    `  `    ``// stores smallest prime factor for every number` `    ``static` `int``[] spf = ``new` `int``[MAXN];` `    `  `    ``// Calculating SPF (Smallest Prime Factor) for every` `    ``// number till MAXN.` `    ``// Time Complexity : O(nloglogn)` `    ``static` `void` `sieve()` `    ``{` `        ``spf[1] = 1;` `        ``for` `(``int` `i = 2; i < MAXN; i++)` `    `  `            ``// marking smallest prime factor for every` `            ``// number to be itself.` `            ``spf[i] = i;` `    `  `        ``// separately marking spf for every even` `        ``// number as 2` `        ``for` `(``int` `i = 4; i < MAXN; i += 2)` `            ``spf[i] = 2;` `    `  `        ``for` `(``int` `i = 3; i * i < MAXN; i++)` `        ``{` `            ``// checking if i is prime` `            ``if` `(spf[i] == i)` `            ``{` `                ``// marking SPF for all numbers divisible by i` `                ``for` `(``int` `j = i * i; j < MAXN; j += i)` `    `  `                    ``// marking spf[j] if it is not ` `                    ``// previously marked` `                    ``if` `(spf[j] == j)` `                        ``spf[j] = i;` `            ``}` `        ``}` `    ``}` `    `  `    ``// A O(log n) function returning primefactorization` `    ``// by dividing by smallest prime factor at every step` `    ``static` `ArrayList getFactorization(``int` `x)` `    ``{` `        ``ArrayList ret = ``new` `ArrayList();` `        ``while` `(x != 1)` `        ``{` `            ``ret.Add(spf[x]);` `            ``x = x / spf[x];` `        ``}` `        ``return` `ret;` `    ``}` `    `  `    ``// Driver code` `    ``public` `static` `void` `Main()` `    ``{` `        ``// precalculating Smallest Prime Factor` `        ``sieve();` `        ``int` `x = 12246;` `        ``Console.Write(``"prime factorization for "` `+ x + ``" : "``);` `    `  `        ``// calling getFactorization function` `        ``ArrayList p = getFactorization(x);` `    `  `        ``for` `(``int` `i = 0; i < p.Count; i++)` `            ``Console.Write(p[i] + ``" "``);` `        ``Console.WriteLine(``""``);` `    ``}` `}`   `// This code is contributed by mits`

## PHP

 ``

## Javascript

 ``

Output:

```prime factorization for 12246 : 2 3 13 157
```

Time Complexity: O(log n), for each query (Time complexity for precomputation is not included)
Auxiliary Space: O(1)

Note : The above code works well for n upto the order of 10^7. Beyond this we will face memory issues.

Time Complexity: The precomputation for smallest prime factor is done in O(n log log n) using sieve. Whereas in the calculation step we are dividing the number every time by the smallest prime number till it becomes 1. So, let’s consider a worst case in which every time the SPF is 2 . Therefore will have log n division steps. Hence, We can say that our Time Complexity will be O(log n) in worst case.

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