Almost Prime Numbers

2

A k-Almost Prime Number is a number having exactly k prime factors (not necessary distinct).

For example,

2, 3, 5, 7, 11 ….(in fact all prime numbers) are 1-Almost Prime Numbers as they have only 1 prime factors (which is themselves).

4, 6, 9…. are 2-Almost Prime Numbers as they have exactly 2 prime factors (4 = 2*2, 6 = 2*3, 9 = 3*3)

Similarly 32 is a 5-Almost Prime Number (32 = 2*2*2*2*2) and so is 72 (2*2*2*3*3)

All the 1-Almost Primes are called as Prime Numbers and all the 2-Almost Prime are called as semi-primes.

The task is to print first n numbers that are k prime.

Examples:

Input : k = 2, n = 5
Output : 4 6 9 10 14
4 has two prime factors, 2 x 2
6 has two prime factors, 2 x 3
Similarly, 9(3 x 3), 10(2 x 5) and 14(2 x 7)

Input : k = 10, n = 2
Output : 1024 1536
1024 and 1536 are first two numbers with 10
prime factors.

We iterate natural numbers and keep printing k-primes till the count of printed k-primes is less than or equal to n. To check if a number is k-prime, we find count of prime factors and if the count is k we consider the number as k-prime. Below is C++ implementation.

// Program to print first n numbers that are k-primes
#include<bits/stdc++.h>

// A function to count all prime factors of a given number
int countPrimeFactors(int n)
{
    int count = 0;

    // Count the number of 2s that divide n
    while (n%2 == 0)
    {
        n = n/2;
        count++;
    }

    // n must be odd at this point. So we can skip one
    // element (Note i = i +2)
    for (int i = 3; i <= sqrt(n); i = i+2)
    {
        // While i divides n, count i and divide n
        while (n%i == 0)
        {
            n = n/i;
            count++;
        }
    }

    // This condition is to handle the case whien n is a
    // prime number greater than 2
    if (n > 2)
        count++;

    return(count);
}

// A function to print the first n numbers that are
// k-almost primes.
void printKAlmostPrimes(int k, int n)
{
    for (int i=1, num=2; i<=n; num++)
    {
        // Print this number if it is k-prime
        if (countPrimeFactors(num) == k)
        {
            printf("%d ", num);

            // Increment count of k-primes printed
            // so far
            i++;
        }
    }
    return;
}

/* Driver program to test above function */
int main()
{
    int n = 10, k = 2;
    printf("First %d %d-almost prime numbers : \n",
           n, k);
    printKAlmostPrimes(k, n);
    return 0;
}

Output :

First 10 2-almost prime numbers : 
4 6 9 10 14 15 21 22 25 26 

References:
https://en.wikipedia.org/wiki/Almost_prime

This article is contributed by Rachit Belwariar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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