Kummer Numbers

Last Updated : 28 Feb, 2022

Given an integer N, the task is to print the first N terms of the Kummer Number.
Examples:

Input: N = 3
Output: 1, 5, 29
1 = -1 + 2
5 = -1 + 2 * 3
29 = -1 + 2 * 3 * 5
Input: N = 5
Output: 1, 5, 29, 209, 2309

A Naive Approach is to check all numbers from 1 to n one by one is prime or not, if yes then store the multiplication in result, similarly store the result of multiplication of primes till n and add -1.
Efficient Approach is to find all the prime up-to n using Sieve of Sundaram and then just calculate the nth kummer number by multiplying them all and adding -1.
Below is the implementation of the above approach:

C++

// C++ program to find Kummer
// number upto n terms
 
#include <bits/stdc++.h>
using namespace std;
const int MAX = 1000000;
 
// vector to store all prime
// less than and equal to 10^6
vector<int> primes;
 
// Function for the Sieve of Sundaram.
// This function stores all
// prime numbers less than MAX in primes
void sieveSundaram()
{
    // In general Sieve of Sundaram,
    // produces primes smaller
    // than (2*x + 2) for a
    // number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half
    // This array is used to separate
    // numbers of the form
    // i+j+2ij from others
    // where 1 <= i <= j
    bool marked[MAX / 2 + 1] = { 0 };
 
    // Main logic of Sundaram.
    // Mark all numbers which
    // do not generate prime number
    // by doing 2*i+1
    for (int i = 1; i <= (sqrt(MAX) - 1) / 2; i++)
 
        for (int j = (i * (i + 1)) << 1;
             j <= MAX / 2;
             j += 2 * i + 1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.push_back(2);
 
    // Print other primes.
    // Remaining primes are of the
    // form 2*i + 1 such that
    // marked[i] is false.
    for (int i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.push_back(2 * i + 1);
}
 
// Function to calculate nth Kummer number
int calculateKummer(int n)
{
    // Multiply first n primes
    int result = 1;
    for (int i = 0; i < n; i++)
        result = result * primes[i];
 
    // return nth Kummer number
    return -1 + result;
}
 
// Driver code
int main()
{
    int n = 5;
    sieveSundaram();
    for (int i = 1; i <= n; i++)
        cout << calculateKummer(i) << ", ";
    return 0;
}

Java

// Java program to find Kummer
// number upto n terms
import java.util.*;
class GFG{
static int MAX = 1000000;
  
// vector to store all prime
// less than and equal to 10^6
static Vector<Integer> primes = new Vector<Integer>();
  
// Function for the Sieve of Sundaram.
// This function stores all
// prime numbers less than MAX in primes
static void sieveSundaram()
{
    // In general Sieve of Sundaram,
    // produces primes smaller
    // than (2*x + 2) for a
    // number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half
    // This array is used to separate
    // numbers of the form
    // i+j+2ij from others
    // where 1 <= i <= j
    boolean marked[] = new boolean[MAX / 2 + 1];
  
    // Main logic of Sundaram.
    // Mark all numbers which
    // do not generate prime number
    // by doing 2*i+1
    for (int i = 1; 
              i <= (Math.sqrt(MAX) - 1) / 2;
             i++)
  
        for (int j = (i * (i + 1)) << 1;
                 j <= MAX / 2;
                 j += 2 * i + 1)
            marked[j] = true;
  
    // Since 2 is a prime number
    primes.add(2);
  
    // Print other primes.
    // Remaining primes are of the
    // form 2*i + 1 such that
    // marked[i] is false.
    for (int i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.add(2 * i + 1);
}
  
// Function to calculate nth Kummer number
static int calculateKummer(int n)
{
    // Multiply first n primes
    int result = 1;
    for (int i = 0; i < n; i++)
        result = result * primes.get(i);
  
    // return nth Kummer number
    return -1 + result;
}
  
// Driver code
public static void main(String[] args)
{
    int n = 5;
    sieveSundaram();
    for (int i = 1; i <= n; i++)
        System.out.print(calculateKummer(i) + ", ");
}
}
 
// This code is contributed by sapnasingh4991

Python3

# Python3 program to find Kummer
# number upto n terms
import math
 
MAX = 1000000
 
# list to store all prime
# less than and equal to 10^6
primes=[]
 
# Function for the Sieve of Sundaram.
# This function stores all
# prime numbers less than MAX in primes
def sieveSundaram():
     
    # In general Sieve of Sundaram,
    # produces primes smaller
    # than (2*x + 2) for a
    # number given number x. Since
    # we want primes smaller than MAX,
    # we reduce MAX to half
    # This list is used to separate
    # numbers of the form
    # i+j+2ij from others
    # where 1 <= i <= j
    marked = [ 0 ] * int(MAX / 2 + 1)
 
    # Main logic of Sundaram.
    # Mark all numbers which
    # do not generate prime number
    # by doing 2*i+1
    for i in range(1, int((math.sqrt(MAX) - 1) // 2) + 1):
        for j in range((i * (i + 1)) << 1,
                        MAX // 2 + 1, 2 * i + 1):
            marked[j] = True
 
    # Since 2 is a prime number
    primes.append(2)
 
    # Print other primes.
    # Remaining primes are of the
    # form 2*i + 1 such that
    # marked[i] is false.
    for i in range(1, MAX // 2 + 1 ):
        if (marked[i] == False):
            primes.append(2 * i + 1)
 
# Function to calculate nth Kummer number
def calculateKummer(n):
 
    # Multiply first n primes
    result = 1
    for i in range(n):
        result = result * primes[i]
 
    # return nth Kummer number
    return (-1 + result)
 
# Driver code
 
# Given N
N = 5
sieveSundaram()
for i in range(1, N + 1):
    print(calculateKummer(i), end = ", ")
     
# This code is contributed by Vishal Maurya

C#

// C# program to find Kummer
// number upto n terms
using System;
using System.Collections.Generic;
 
class GFG{
static int MAX = 1000000;
 
// vector to store all prime
// less than and equal to 10^6
static List<int> primes = new List<int>();
 
// Function for the Sieve of Sundaram.
// This function stores all
// prime numbers less than MAX in primes
static void sieveSundaram()
{
    // In general Sieve of Sundaram,
    // produces primes smaller
    // than (2*x + 2) for a
    // number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half
    // This array is used to separate
    // numbers of the form
    // i+j+2ij from others
    // where 1 <= i <= j
    bool []marked = new bool[MAX / 2 + 1];
 
    // Main logic of Sundaram.
    // Mark all numbers which
    // do not generate prime number
    // by doing 2*i+1
    for (int i = 1; 
             i <= (Math.Sqrt(MAX) - 1) / 2;
             i++)
 
        for (int j = (i * (i + 1)) << 1;
                 j <= MAX / 2;
                 j += 2 * i + 1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.Add(2);
 
    // Print other primes.
    // Remaining primes are of the
    // form 2*i + 1 such that
    // marked[i] is false.
    for (int i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.Add(2 * i + 1);
}
 
// Function to calculate nth Kummer number
static int calculateKummer(int n)
{
    // Multiply first n primes
    int result = 1;
    for (int i = 0; i < n; i++)
        result = result * primes[i];
 
    // return nth Kummer number
    return -1 + result;
}
 
// Driver code
public static void Main(String[] args)
{
    int n = 5;
    sieveSundaram();
    for (int i = 1; i <= n; i++)
        Console.Write(calculateKummer(i) + ", ");
}
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// JavaScript program to find Kummer
// number upto n terms
 
// Function to count the number of 
// ways to decode the given digit 
// sequence 
let MAX = 1000000;
    
// vector to store all prime
// less than and equal to 10^6
let primes = [];
    
// Function for the Sieve of Sundaram.
// This function stores all
// prime numbers less than MAX in primes
function sieveSundaram()
{
    // In general Sieve of Sundaram,
    // produces primes smaller
    // than (2*x + 2) for a
    // number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half
    // This array is used to separate
    // numbers of the form
    // i+j+2ij from others
    // where 1 <= i <= j
    let marked = Array.from(
    {length: MAX / 2 + 1}, (_, i) => 0); 
    
    // Main logic of Sundaram.
    // Mark all numbers which
    // do not generate prime number
    // by doing 2*i+1
    for (let i = 1; 
              i <= (Math.sqrt(MAX) - 1) / 2;
             i++)
    
        for (let j = (i * (i + 1)) << 1;
                 j <= MAX / 2;
                 j += 2 * i + 1)
            marked[j] = true;
    
    // Since 2 is a prime number
    primes.push(2);
    
    // Print other primes.
    // Remaining primes are of the
    // form 2*i + 1 such that
    // marked[i] is false.
    for (let i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.push(2 * i + 1);
}
    
// Function to calculate nth Kummer number
function calculateKummer(n)
{
    // Multiply first n primes
    let result = 1;
    for (let i = 0; i < n; i++)
        result = result * primes[i];
    
    // return nth Kummer number
    return -1 + result;
}
// Driver Code
 
        let n = 5;
    sieveSundaram();
    for (let i = 1; i <= n; i++)
        document.write(calculateKummer(i) + ", ");
           
</script>