# Find the Nth term of the Zumkeller Numbers

• Last Updated : 26 Sep, 2022

Zumkeller numbers are the set of numbers whose divisors can be partitioned into two disjoint sets that sum to the same value. The first few zumkeller numbers are 6, 12, 20, 24, 28, 30, 40, 42, 48, 54, …
In this article, we will find the Nth Zumkeller number.
Find the Nth Zumkeller Number: Given a number N, the task is to find the Nth Zumkeller number.
Examples:

Input: N = 2
Output: 12
Explanation:
The second Zumkeller number is 12.
Input: N = 5
Output: 28

Approach: The following steps are followed to compute the answer.

1. Get the number N.
2. Iterate over the loop starting from i = 1 until Nth Zumkeller number is found.
3. Check the number ‘i’ is a zumkeller number or not.
4. If yes, then repeat the above step for i+1 and increment the counter.
5. If no, then repeat the above step for i+1 without incrementing the counter.
6. Finally, when the counter is equal to the number N, that number is the N-th zumkeller number. Print the value of ‘i’ and break the loop.

Below is the implementation of the above approach:

## Python3

 `# Python program to find the``# N-th Zumkeller number` `# Function to find all the``# divisors``def` `Divisors(n) :``    ``l ``=` `[]``    ``i ``=` `1``    ``while` `i <``=` `n :``        ``if` `(n ``%` `i ``=``=` `0``) :``            ``l.append(i)``        ``i ``=` `i ``+` `1``    ``return` `l` `# Function to check if the sum``# of the subset of divisors is``# equal to sum / 2 or not``def` `PowerSet(arr, n, s):``       ` `    ``# List to find all the``    ``# subsets of the given set.``    ``# Any repeated subset is``    ``# considered only ``    ``# once in the output``    ``_list ``=` `[]``   ` `    ``# Run a counter i ``    ``for` `i ``in` `range``(``2``*``*``n):``        ``subset ``=` `""``   ` `        ``# Consider each element``        ``# in the set``        ``for` `j ``in` `range``(n):``   ` `            ``# Check if j-th bit in``            ``# the i is set. ``            ``# If the bit is set,``            ``# we consider ``            ``# j-th element from set``            ``if` `(i & (``1` `<< j)) !``=` `0``:``                ``subset ``+``=` `str``(arr[j]) ``+` `"|"``   ` `        ``# Check if the subset is``        ``# encountered for the first time.``        ``if` `subset ``not` `in` `_list ``and` `len``(subset) > ``0``:``            ``_list.append(subset)``   ` `    ``# Consider every subset``    ``for` `subset ``in` `_list:``        ``sum` `=` `0` `        ``# Split the subset and``        ``# sum of its elements``        ``arr ``=` `subset.split(``'|'``)``        ``for` `string ``in` `arr[:``-``1``]:``            ``sum` `+``=` `int``(string)` `            ``# If the sum is equal``            ``# to S``            ``if` `sum` `=``=` `s:``                ``return` `True` `    ``return` `False` `# Function to check if a number``# is a Zumkeller number``def` `isZumkeller(n):` `    ``# To find all the divisors``    ``# of a number N``    ``d ``=` `Divisors(n)` `    ``# Finding the sum of``    ``# all the divisors``    ``s ``=` `sum``(d)` `    ``# Check for the condition that``    ``# sum must be even and the``    ``# maximum divisor is less than``    ``# or equal to sum / 2.``    ``# If the sum is odd and the``    ``# maximum divisor is greater than``    ``# sum / 2, then it is not possible``    ``# to divide the divisors into``    ``# two sets``    ``if` `not` `s ``%` `2` `and` `max``(d) <``=` `s ``/` `2``:` `        ``# For all the subsets of``        ``# the divisors``        ``if` `PowerSet(d, ``len``(d), s ``/` `2``) :``                ``return` `True` `    ``return` `False` ` ` `# Function to print N-th``# Zumkeller number``def` `printZumkellers(N):``    ``val ``=` `0``    ``ans ``=` `0` `    ``# Iterating through all``    ``# the numbers``    ``for` `n ``in` `range``(``1``, ``10``*``*``5``):`` ` `        ``# Check if n is a``        ``# Zumkeller number``        ``if` `isZumkeller(n):``            ``ans ``=` `n``            ``val ``+``=` `1` `            ``# Check if N-th Zumkeller number``            ``# is obtained or not``            ``if` `val >``=` `N:``                ``break`               `    ``print``(ans)`` ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:`` ` `    ``N ``=` `4`` ` `    ``printZumkellers(N)`

## Javascript

 `// Python program to find the``// N-th Zumkeller number` `// Function to find all the``// divisors``function` `Divisors(n)``{``    ``let l = []``    ``let i = 1``    ``while` `(i <= n)``    ``{``        ``if` `(n % i == 0)``            ``l.push(i)``        ``i = i + 1``    ``}``    ``return` `l``}` `// Function to check if the sum``// of the subset of divisors is``// equal to sum / 2 or not``function` `PowerSet(arr, n, s)``{``       ` `    ``// List to find all the``    ``// subsets of the given set.``    ``// Any repeated subset is``    ``// considered only ``    ``// once in the output``    ``let _list = []``   ` `    ``// Run a counter i ``    ``for` `(``var` `i = 0; i < 2 ** n; i++)``    ``{``        ``let subset = ``""``   ` `        ``// Consider each element``        ``// in the set``        ``for` `(``var` `j = 0; j < n; j++)``        ``{``   ` `            ``// Check if j-th bit in``            ``// the i is set. ``            ``// If the bit is set,``            ``// we consider ``            ``// j-th element from set``            ``if` `((i & (1 << j)) != 0)``                ``subset += ((arr[j]) + ``"|"``)``        ``}``   ` `        ``// Check if the subset is``        ``// encountered for the first time.``        ``if` `(!_list.includes(subset) && subset.length > 0)``            ``_list.push(subset)``    ``}``   ` `    ``// Consider every subset``    ``for` `(let subset of _list)``    ``{``        ``let sum = 0` `        ``// Split the subset and``        ``// sum of its elements``        ``let arr = subset.split(``'|'``)``        ``arr.pop()``        ``for` `(let string of arr)``        ``{``            ``sum += parseInt(string)` `            ``// If the sum is equal``            ``// to S``            ``if` `(sum == s)``                ``return` `true``        ``}``    ``}``    ``return` `false``}` `// Function to check if a number``// is a Zumkeller number``function` `isZumkeller(n)``{` `    ``// To find all the divisors``    ``// of a number N``    ``let d = Divisors(n)` `    ``// Finding the sum of``    ``// all the divisors``    ``let s = 0;``    ``for` `(let d1 of d)``        ``s += d1;`  `    ``// Check for the condition that``    ``// sum must be even and the``    ``// maximum divisor is less than``    ``// or equal to sum / 2.``    ``// If the sum is odd and the``    ``// maximum divisor is greater than``    ``// sum / 2, then it is not possible``    ``// to divide the divisors into``    ``// two sets``    ``let maxd = Math.max(...d)``    ``if` `(s % 2 == 0  && maxd <= (s / 2) )` `        ``// For all the subsets of``        ``// the divisors``        ``if` `(PowerSet(d, d.length, s / 2))``                ``return` `true` `    ``return` `false``}` `// Function to print N-th``// Zumkeller number``function` `printZumkellers(N)``{``    ``let val = 0``    ``let ans = 0` `    ``// Iterating through all``    ``// the numbers``    ``for` `(``var` `n = 1; n < 10 ** 5; n ++)``    ``{`` ` `        ``// Check if n is a``        ``// Zumkeller number``        ``if` `(isZumkeller(n))``        ``{``            ``ans = n``            ``val += 1` `            ``// Check if N-th Zumkeller number``            ``// is obtained or not``            ``if` `(val >= N)``                ``break`   `        ``}``    ``}` `    ``console.log(ans)``}`` ` `// Driver code``let N = 4`` ` `printZumkellers(N)` `// This code is contributed by phasing17`

Output:

`24`

My Personal Notes arrow_drop_up