Print numbers such that no two consecutive numbers are co-prime and every three consecutive numbers are co-prime

Given an integer N, the task is to print N integers ≤ 109 such that no two consecutive of these integers are co-prime and every 3 consecutive are co-prime.
Examples:  

Input: N = 3 
Output: 6 15 10
Input: N = 4 
Output: 6 15 35 14 
 

Approach:  

  • We can just multiply consecutive primes and for the last number just multiply the gcd(last, last-1) * 2. We do this so that the (n – 1)th number, nth and 1st number can also follow the property mentioned in the problem statement.
  • Another important part of the problem is the fact that the numbers should be ≤ 109. If you just multiply consecutive prime numbers, after around 3700 numbers the value will cross 109. So we need to only use those prime numbers whose product won’t cross 109.
  • To do this efficiently, consider a small number of primes say the first 550 primes, and select them in a way such that on making a product no number gets repeated. We first choose every prime consecutively and then choose the primes with an interval of 2 and then 3 and so on, doing that we already make sure that no number gets repeated.

So we will select 
5, 11, 17, … 
Next time, we can start with 7 and select, 
7, 13, 19, … 
 

Below is the implementation of the above approach: 
 



C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
#define limit 1000000000
#define MAX_PRIME 2000000
#define MAX 1000000
#define I_MAX 50000
 
map<int, int> mp;
 
int b[MAX];
int p[MAX];
int j = 0;
bool prime[MAX_PRIME + 1];
 
// Function to generate Sieve of
// Eratosthenes
void SieveOfEratosthenes(int n)
{
    memset(prime, true, sizeof(prime));
 
    for (int p = 2; p * p <= n; p++) {
 
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true) {
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    // Add the prime numbers to the array b
    for (int p = 2; p <= n; p++) {
        if (prime[p]) {
            b[j++] = p;
        }
    }
}
 
// Function to return the gcd of a and b
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// Function to print the required
// sequence of integers
void printSeries(int n)
{
    SieveOfEratosthenes(MAX_PRIME);
 
    int i, g, k, l, m, d;
    int ar[I_MAX + 2];
 
    for (i = 0; i < j; i++) {
        if ((b[i] * b[i + 1]) > limit)
            break;
 
        // Including the primes in a series
        // of primes which will be later
        // multiplied
        p[i] = b[i];
 
        // This is done to mark a product
        // as existing
        mp[b[i] * b[i + 1]] = 1;
    }
 
    // Maximum number of primes that we consider
    d = 550;
    bool flag = false;
 
    // For different interval
    for (k = 2; (k < d - 1) && !flag; k++) {
 
        // For different starting index of jump
        for (m = 2; (m < d) && !flag; m++) {
 
            // For storing the numbers
            for (l = m + k; l < d; l += k) {
 
                // Checking for occurrence of a
                // product. Also checking for the
                // same prime occurring consecutively
                if (((b[l] * b[l + k]) < limit)
                    && (l + k) < d && p[i - 1] != b[l + k]
                    && p[i - 1] != b[l] && mp[b[l] * b[l + k]] != 1) {
                    if (mp[p[i - 1] * b[l]] != 1) {
 
                        // Including the primes in a
                        // series of primes which will
                        // be later multiplied
                        p[i] = b[l];
                        mp[p[i - 1] * b[l]] = 1;
                        i++;
                    }
                }
 
                if (i >= I_MAX) {
                    flag = true;
                    break;
                }
            }
        }
    }
 
    for (i = 0; i < n; i++)
        ar[i] = p[i] * p[i + 1];
 
    for (i = 0; i < n - 1; i++)
        cout << ar[i] << " ";
 
    g = gcd(ar[n - 1], ar[n - 2]);
    cout << g * 2 << endl;
}
 
// Driver Code
int main()
{
    int n = 4;
 
    printSeries(n);
 
    return 0;
}

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Java

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// Java implementation of the approach
import java.util.*;
 
class GFG
{
 
static int limit = 1000000000;
static int MAX_PRIME = 2000000;
static int MAX = 1000000;
static int I_MAX = 50000;
 
static HashMap<Integer,
               Integer> mp = new HashMap<Integer,
                                         Integer>();
 
static int []b = new int[MAX];
static int []p = new int[MAX];
static int j = 0;
static boolean []prime = new boolean[MAX_PRIME + 1];
 
// Function to generate Sieve of
// Eratosthenes
static void SieveOfEratosthenes(int n)
{
    for(int i = 0; i < MAX_PRIME + 1; i++)
        prime[i] = true;
 
    for (int p = 2; p * p <= n; p++)
    {
 
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true)
        {
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    // Add the prime numbers to the array b
    for (int p = 2; p <= n; p++)
    {
        if (prime[p])
        {
            b[j++] = p;
        }
    }
}
 
// Function to return the gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// Function to print the required
// sequence of integers
static void printSeries(int n)
{
    SieveOfEratosthenes(MAX_PRIME);
 
    int i, g, k, l, m, d;
    int []ar = new int[I_MAX + 2];
 
    for (i = 0; i < j; i++)
    {
        if ((b[i] * b[i + 1]) > limit)
            break;
 
        // Including the primes in a series
        // of primes which will be later
        // multiplied
        p[i] = b[i];
 
        // This is done to mark a product
        // as existing
        mp.put(b[i] * b[i + 1], 1);
    }
 
    // Maximum number of primes that we consider
    d = 550;
    boolean flag = false;
 
    // For different interval
    for (k = 2; (k < d - 1) && !flag; k++)
    {
 
        // For different starting index of jump
        for (m = 2; (m < d) && !flag; m++)
        {
 
            // For storing the numbers
            for (l = m + k; l < d; l += k)
            {
 
                // Checking for occurrence of a
                // product. Also checking for the
                // same prime occurring consecutively
                if (((b[l] * b[l + k]) < limit) &&
                      mp.containsKey(b[l] * b[l + k]) &&
                      mp.containsKey(p[i - 1] * b[l]) &&
                      (l + k) < d && p[i - 1] != b[l + k] &&
                                         p[i - 1] != b[l] &&
                             mp.get(b[l] * b[l + k]) != 1)
                    {
                    if (mp.get(p[i - 1] * b[l]) != 1)
                    {
 
                        // Including the primes in a
                        // series of primes which will
                        // be later multiplied
                        p[i] = b[l];
                        mp.put(p[i - 1] * b[l], 1);
                        i++;
                    }
                }
 
                if (i >= I_MAX)
                {
                    flag = true;
                    break;
                }
            }
        }
    }
 
    for (i = 0; i < n; i++)
        ar[i] = p[i] * p[i + 1];
 
    for (i = 0; i < n - 1; i++)
        System.out.print(ar[i]+" ");
 
    g = gcd(ar[n - 1], ar[n - 2]);
    System.out.print(g * 2);
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 4;
    printSeries(n);
}
}
 
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation of
# the above approach
limit = 1000000000
MAX_PRIME = 2000000
MAX = 1000000
I_MAX = 50000
 
mp = {}
 
b = [0] * MAX
p = [0] * MAX
j = 0
prime = [True] * (MAX_PRIME + 1)
 
# Function to generate Sieve of
# Eratosthenes
def SieveOfEratosthenes(n):
    global j
    p = 2
    while p * p <= n:
 
        # If prime[p] is not changed,
        # then it is a prime
        if (prime[p] == True):
            for i in range(p * p, n + 1, p):
                prime[i] = False
        p += 1
 
    # Add the prime numbers to the array b
    for p in range(2, n + 1):
        if (prime[p]):
            b[j] = p
            j += 1
 
# Function to return
# the gcd of a and b
def gcd(a, b):
 
    if (b == 0):
        return a
    return gcd(b, a % b)
 
# Function to print the required
# sequence of integers
def printSeries(n):
 
    SieveOfEratosthenes(MAX_PRIME)
 
    ar = [0] * (I_MAX + 2)
 
    for i in range(j):
        if ((b[i] * b[i + 1]) > limit):
            break
 
        # Including the primes in a series
        # of primes which will be later
        # multiplied
        p[i] = b[i]
 
        # This is done to mark a product
        # as existing
        mp[b[i] * b[i + 1]] = 1
 
    # Maximum number of
    # primes that we consider
    d = 550
    flag = False
 
    # For different interval
    k = 2
    while (k < d - 1) and not flag:
 
        # For different starting
        # index of jump
        m = 2
        while (m < d) and not flag:
 
            # For storing the numbers
            for l in range(m + k, d, k):
 
                # Checking for occurrence of a
                # product. Also checking for the
                # same prime occurring consecutively
                if (((b[l] * b[l + k]) < limit) and
                    (l + k) < d and p[i - 1] != b[l + k] and
                     p[i - 1] != b[l] and
                     ((b[l] * b[l + k] in mp) and
                     mp[b[l] * b[l + k]] != 1)):
                   
                    if (mp[p[i - 1] * b[l]] != 1):
 
                        # Including the primes in a
                        # series of primes which will
                        # be later multiplied
                        p[i] = b[l]
                        mp[p[i - 1] * b[l]] = 1
                        i += 1
 
                if (i >= I_MAX):
                    flag = True
                    break
            m += 1
        k += 1
 
    for i in range(n):
        ar[i] = p[i] * p[i + 1]
 
    for i in range(n - 1):
        print(ar[i], end = " ")
 
    g = gcd(ar[n - 1], ar[n - 2])
    print(g * 2)
 
# Driver Code
if __name__ == "__main__":
    n = 4
    printSeries(n)
 
# This code is contributed by Chitranayal

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C#

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// C# implementation of the approach
using System;
using System.Collections.Generic;            
     
class GFG
{
 
static int limit = 1000000000;
static int MAX_PRIME = 2000000;
static int MAX = 1000000;
static int I_MAX = 50000;
 
static Dictionary<int,
                  int> mp = new Dictionary<int,
                                           int>();
 
static int []b = new int[MAX];
static int []p = new int[MAX];
static int j = 0;
static bool []prime = new bool[MAX_PRIME + 1];
 
// Function to generate Sieve of
// Eratosthenes
static void SieveOfEratosthenes(int n)
{
    for(int i = 0; i < MAX_PRIME + 1; i++)
        prime[i] = true;
 
    for (int p = 2; p * p <= n; p++)
    {
 
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true)
        {
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    // Add the prime numbers to the array b
    for (int p = 2; p <= n; p++)
    {
        if (prime[p])
        {
            b[j++] = p;
        }
    }
}
 
// Function to return the gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// Function to print the required
// sequence of integers
static void printSeries(int n)
{
    SieveOfEratosthenes(MAX_PRIME);
 
    int i, g, k, l, m, d;
    int []ar = new int[I_MAX + 2];
 
    for (i = 0; i < j; i++)
    {
        if ((b[i] * b[i + 1]) > limit)
            break;
 
        // Including the primes in a series
        // of primes which will be later
        // multiplied
        p[i] = b[i];
 
        // This is done to mark a product
        // as existing
        mp.Add(b[i] * b[i + 1], 1);
    }
 
    // Maximum number of primes that we consider
    d = 550;
    bool flag = false;
 
    // For different interval
    for (k = 2; (k < d - 1) && !flag; k++)
    {
 
        // For different starting index of jump
        for (m = 2; (m < d) && !flag; m++)
        {
 
            // For storing the numbers
            for (l = m + k; l < d; l += k)
            {
 
                // Checking for occurrence of a
                // product. Also checking for the
                // same prime occurring consecutively
                if (((b[l] * b[l + k]) < limit) &&
                    mp.ContainsKey(b[l] * b[l + k]) &&
                    mp.ContainsKey(p[i - 1] * b[l]) &&
                    (l + k) < d && p[i - 1] != b[l + k] &&
                                       p[i - 1] != b[l] &&
                            mp[b[l] * b[l + k]] != 1)
                    {
                    if (mp[p[i - 1] * b[l]] != 1)
                    {
 
                        // Including the primes in a
                        // series of primes which will
                        // be later multiplied
                        p[i] = b[l];
                        mp.Add(p[i - 1] * b[l], 1);
                        i++;
                    }
                }
 
                if (i >= I_MAX)
                {
                    flag = true;
                    break;
                }
            }
        }
    }
 
    for (i = 0; i < n; i++)
        ar[i] = p[i] * p[i + 1];
 
    for (i = 0; i < n - 1; i++)
        Console.Write(ar[i] + " ");
 
    g = gcd(ar[n - 1], ar[n - 2]);
    Console.Write(g * 2);
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 4;
    printSeries(n);
}
}
 
// This code is contributed by 29AjayKumar

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Output: 

6 15 35 14


 

Another approach: List all the prime numbers up to 6 million by using the Sieve of Eratosthenes. We know the base condition i.e. N = 3 forms {6, 10, 15}. 
So, we use these three values to generate further terms of the sequence. 
As {2, 3, 5}, these primes can not be used to generate sequences because they are already used in {6, 10, 15}. We also can’t use {7, 11} which we’ll see later. 
Now we have a prime list {13, 17, 19, 23, 29, ……}. We take the first prime and multiply it with 
6, second with 15, third with 10, again 4th with 6, and so on… 

13 * 6, 17 * 15, 19 * 10, 23 * 6, 29 * 15, ........upto N - 2 terms.
(N - 1)th term = (N - 1)th prime * 7.
Nth term = 7 * 11.
again, first term = first term * 11 to make 1st and last noncoprime.
For example, N = 5
6 * 11 * 13, 15 * 17, 10 * 19, 11 * 19, 7 * 11


Now we see that we can not use 7 and 11 from the list as these are used to generate last and second last term.
Below is the implementation of the above approach:  

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
const int MAX = 620000;
int prime[MAX];
 
// Function for Sieve of Eratosthenes
void Sieve()
{
    for (int i = 2; i < MAX; i++) {
        if (prime[i] == 0) {
            for (int j = 2 * i; j < MAX; j += i) {
                prime[j] = 1;
            }
        }
    }
}
 
// Function to print the required sequence
void printSequence(int n)
{
    Sieve();
    vector<int> v, u;
 
    // Store only the required primes
    for (int i = 13; i < MAX; i++) {
        if (prime[i] == 0) {
            v.push_back(i);
        }
    }
    // Base condition
    if (n == 3) {
        cout << 6 << " " << 10 << " " << 15;
        return;
    }
 
    int k;
    for (k = 0; k < n - 2; k++) {
 
        // First integer in the list
        if (k % 3 == 0) {
            u.push_back(v[k] * 6);
        }
 
        // Second integer in the list
        else if (k % 3 == 1) {
 
            u.push_back(v[k] * 15);
        }
 
        // Third integer in the list
        else {
            u.push_back(v[k] * 10);
        }
    }
    k--;
 
    // Generate (N - 1)th term
    u.push_back(v[k] * 7);
 
    // Generate Nth term
    u.push_back(7 * 11);
 
    // Modify first term
    u[0] = u[0] * 11;
 
    // Print the sequence
    for (int i = 0; i < u.size(); i++) {
        cout << u[i] << " ";
    }
}
 
// Driver code
int main()
{
    int n = 4;
    printSequence(n);
 
    return 0;
}

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// Java implementation of the approach
import java.util.*;
 
class GFG
{
    static int MAX = 620000;
    static int[] prime = new int[MAX];
 
    // Function for Sieve of Eratosthenes
    static void Sieve()
    {
        for (int i = 2; i < MAX; i++)
        {
            if (prime[i] == 0)
            {
                for (int j = 2 * i;
                         j < MAX; j += i)
                {
                    prime[j] = 1;
                }
            }
        }
    }
 
    // Function to print the required sequence
    static void printSequence(int n)
    {
        Sieve();
        Vector<Integer> v = new Vector<Integer>();
        Vector<Integer> u = new Vector<Integer>();
 
        // Store only the required primes
        for (int i = 13; i < MAX; i++)
        {
            if (prime[i] == 0)
            {
                v.add(i);
            }
        }
         
        // Base condition
        if (n == 3)
        {
            System.out.print(6 + " " + 10 + " " + 15);
            return;
        }
 
        int k;
        for (k = 0; k < n - 2; k++)
        {
 
            // First integer in the list
            if (k % 3 == 0)
            {
                u.add(v.get(k) * 6);
            }
             
            // Second integer in the list
            else if (k % 3 == 1)
            {
 
                u.add(v.get(k) * 15);
            }
             
            // Third integer in the list
            else
            {
                u.add(v.get(k) * 10);
            }
        }
        k--;
 
        // Generate (N - 1)th term
        u.add(v.get(k) * 7);
 
        // Generate Nth term
        u.add(7 * 11);
 
        // Modify first term
        u.set(0, u.get(0) * 11);
 
        // Print the sequence
        for (int i = 0; i < u.size(); i++)
        {
            System.out.print(u.get(i) + " ");
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 4;
        printSequence(n);
    }
}
 
// This code is contributed by Rajput-Ji

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Python3

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# Python3 program for the above approach
MAX = 620000
prime = [0] * MAX
 
# Function for Sieve of Eratosthenes
def Sieve():
 
    for i in range(2, MAX):
        if (prime[i] == 0):
            for j in range(2 * i, MAX, i):
                prime[j] = 1
 
# Function to print the required sequence
def printSequence (n):
 
    Sieve()
    v = []
    u = []
 
    # Store only the required primes
    for i in range(13, MAX):
        if (prime[i] == 0):
            v.append(i)
 
    # Base condition
    if (n == 3):
        print(6, 10, 15)
        return
 
    k = 0
    for k in range(n - 2):
 
        # First integer in the list
        if (k % 3 == 0):
            u.append(v[k] * 6)
 
        # Second integer in the list
        elif (k % 3 == 1):
            u.append(v[k] * 15)
 
        # Third integer in the list
        else:
            u.append(v[k] * 10)
     
    # Generate (N - 1)th term
    u.append(v[k] * 7)
 
    # Generate Nth term
    u.append(7 * 11)
 
    # Modify first term
    u[0] = u[0] * 11
 
    # Print the sequence
    print(*u)
 
# Driver code
if __name__ == '__main__':
 
    n = 4
    printSequence(n)
 
# This code is contributed by himanshu77

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C#

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// C# implementation of the approach
using System;
using System.Collections.Generic;
 
class GFG
{
    static int MAX = 620000;
    static int[] prime = new int[MAX];
 
    // Function for Sieve of Eratosthenes
    static void Sieve()
    {
        for (int i = 2; i < MAX; i++)
        {
            if (prime[i] == 0)
            {
                for (int j = 2 * i;
                        j < MAX; j += i)
                {
                    prime[j] = 1;
                }
            }
        }
    }
 
    // Function to print the required sequence
    static void printSequence(int n)
    {
        Sieve();
        List<int> v = new List<int>();
        List<int> u = new List<int>();
 
        // Store only the required primes
        for (int i = 13; i < MAX; i++)
        {
            if (prime[i] == 0)
            {
                v.Add(i);
            }
        }
         
        // Base condition
        if (n == 3)
        {
            Console.Write(6 + " " + 10 + " " + 15);
            return;
        }
 
        int k;
        for (k = 0; k < n - 2; k++)
        {
 
            // First integer in the list
            if (k % 3 == 0)
            {
                u.Add(v[k] * 6);
            }
             
            // Second integer in the list
            else if (k % 3 == 1)
            {
 
                u.Add(v[k] * 15);
            }
             
            // Third integer in the list
            else
            {
                u.Add(v[k] * 10);
            }
        }
        k--;
 
        // Generate (N - 1)th term
        u.Add(v[k] * 7);
 
        // Generate Nth term
        u.Add(7 * 11);
 
        // Modify first term
        u[0] = u[0] * 11;
 
        // Print the sequence
        for (int i = 0; i < u.Count; i++)
        {
            Console.Write(u[i] + " ");
        }
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int n = 4;
        printSequence(n);
    }
}
 
// This code is contributed by Princi Singh

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Output: 

858 255 119 77


 

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