Least prime factor of numbers till n
Given a number n, print least prime factors of all numbers from 1 to n. The least prime factor of an integer n is the smallest prime number that divides the number. The least prime factor of all even numbers is 2. A prime number is its own least prime factor (as well as its own greatest prime factor).
Note: We need to print 1 for 1.
Input : 6 Output : Least Prime factor of 1: 1 Least Prime factor of 2: 2 Least Prime factor of 3: 3 Least Prime factor of 4: 2 Least Prime factor of 5: 5 Least Prime factor of 6: 2
We can use a variation of sieve of Eratosthenes to solve the above problem.
- Create a list of consecutive integers from 2 through n: (2, 3, 4, …, n).
- Initially, let i equal 2, the smallest prime number.
- Enumerate the multiples of i by counting to n from 2i in increments of i, and mark them as having least prime factor as i (if not already marked). Also mark i as least prime factor of i (i itself is a prime number).
- Find the first number greater than i in the list that is not marked. If there was no such number, stop. Otherwise, let i now equal this new number (which is the next prime), and repeat from step 3.
Below is the implementation of the algorithm, where least_prime saves the value of the least prime factor corresponding to the respective index.
Least Prime factor of 1: 1 Least Prime factor of 2: 2 Least Prime factor of 3: 3 Least Prime factor of 4: 2 Least Prime factor of 5: 5 Least Prime factor of 6: 2 Least Prime factor of 7: 7 Least Prime factor of 8: 2 Least Prime factor of 9: 3 Least Prime factor of 10: 2
Time Complexity: O(nlog(n))
Auxiliary Space: O(n)
Can we extend this algorithm or use least_prime to find all the prime factors for numbers till n?
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