In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
Given a number N, the task is to check if the number is weird or not.
Examples:
Input: 40
Output: The number is not weird
1+4+5+10+20=40, hence it is not weird.Input: 70
Output: The number is Weird
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70.
The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2+4+6 = 12. The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, …
Approach: Check if the number is abundant or not. The approach has been discussed here. Once the checking has been done, check if the number is semiperfect or not. The approach for checking semiperfect numbers has been discussed here.
Below is the implementation of the above approach:
// C++ program to check if the // number is weird or not #include <bits/stdc++.h> using namespace std;
// code to find all the factors of // the number excluding the number itself vector< int > factors( int n)
{ // vector to store the factors
vector< int > v;
v.push_back(1);
// note that this loop runs till sqrt(n)
for ( int i = 2; i <= sqrt (n); i++) {
// if the value of i is a factor
if (n % i == 0) {
v.push_back(i);
// condition to check the
// divisor is not the number itself
if (n / i != i) {
v.push_back(n / i);
}
}
}
// return the vector
return v;
} // Function to check if the number // is abundant or not bool checkAbundant( int n)
{ vector< int > v;
int sum = 0;
// find the divisors using function
v = factors(n);
// sum all the factors
for ( int i = 0; i < v.size(); i++) {
sum += v[i];
}
// check for abundant or not
if (sum > n)
return true ;
else
return false ;
} // Function to check if the // number is semi-perfect or not bool checkSemiPerfect( int n)
{ vector< int > v;
// find the divisors
v = factors(n);
// sorting the vector
sort(v.begin(), v.end());
int r = v.size();
// subset to check if no is semiperfect
bool subset[r + 1][n + 1];
// initialising 1st column to true
for ( int i = 0; i <= r; i++)
subset[i][0] = true ;
// initialising 1st row except zero position to 0
for ( int i = 1; i <= n; i++)
subset[0][i] = false ;
// loop to find whether the number is semiperfect
for ( int i = 1; i <= r; i++) {
for ( int j = 1; j <= n; j++) {
// calculation to check if the
// number can be made by summation of divisors
if (j < v[i - 1])
subset[i][j] = subset[i - 1][j];
else {
subset[i][j] = subset[i - 1][j] ||
subset[i - 1][j - v[i - 1]];
}
}
}
// if not possible to make the
// number by any combination of divisors
if ((subset[r][n]) == 0)
return false ;
else
return true ;
} // Function to check for // weird or not bool checkweird( int n)
{ if (checkAbundant(n) == true &&
checkSemiPerfect(n) == false )
return true ;
else
return false ;
} // Driver Code int main()
{ int n = 70;
if (checkweird(n))
cout << "Weird Number" ;
else
cout << "Not Weird Number" ;
return 0;
} |
// Java program to check if // the number is weird or not import java.util.*;
class GFG
{ // code to find all the // factors of the number // excluding the number itself static ArrayList<Integer> factors( int n)
{ // ArrayList to store
// the factors
ArrayList<Integer> v = new ArrayList<Integer>();
v.add( 1 );
// note that this loop
// runs till sqrt(n)
for ( int i = 2 ;
i <= Math.sqrt(n); i++)
{
// if the value of
// i is a factor
if (n % i == 0 )
{
v.add(i);
// condition to check
// the divisor is not
// the number itself
if (n / i != i)
{
v.add(n / i);
}
}
}
// return the ArrayList
return v;
} // Function to check if the // number is abundant or not static boolean checkAbundant( int n)
{ ArrayList<Integer> v;
int sum = 0 ;
// find the divisors
// using function
v = factors(n);
// sum all the factors
for ( int i = 0 ; i < v.size(); i++)
{
sum += v.get(i);
}
// check for abundant
// or not
if (sum > n)
return true ;
else
return false ;
} // Function to check if the // number is semi-perfect or not static boolean checkSemiPerfect( int n)
{ ArrayList<Integer> v;
// find the divisors
v = factors(n);
// sorting the ArrayList
Collections.sort(v);
int r = v.size();
// subset to check if
// no is semiperfect
boolean subset[][] = new boolean [r + 1 ][n + 1 ];
// initialising 1st
// column to true
for ( int i = 0 ; i <= r; i++)
subset[i][ 0 ] = true ;
// initialising 1st row except
// zero position to 0
for ( int i = 1 ; i <= n; i++)
subset[ 0 ][i] = false ;
// loop to find whether
// the number is semiperfect
for ( int i = 1 ; i <= r; i++)
{
for ( int j = 1 ; j <= n; j++)
{
// calculation to check
// if the number can be
// made by summation of
// divisors
if (j < v.get(i - 1 ))
subset[i][j] = subset[i - 1 ][j];
else {
subset[i][j] = subset[i - 1 ][j] ||
subset[i - 1 ][j -
v.get(i - 1 )];
}
}
}
// if not possible to make
// the number by any
// combination of divisors
if ((subset[r][n]) == false )
return false ;
else
return true ;
} // Function to check // for weird or not static boolean checkweird( int n)
{ if (checkAbundant(n) == true &&
checkSemiPerfect(n) == false )
return true ;
else
return false ;
} // Driver Code public static void main(String args[])
{ int n = 70 ;
if (checkweird(n))
System.out.println( "Weird Number" );
else
System.out.println( "Not Weird Number" );
} } // This code is contributed // by Arnab Kundu |
# Python 3 program to check if the # number is weird or not from math import sqrt
# code to find all the factors of # the number excluding the number itself def factors(n):
# vector to store the factors
v = []
v.append( 1 )
# note that this loop runs till sqrt(n)
for i in range ( 2 , int (sqrt(n)) + 1 , 1 ):
# if the value of i is a factor
if (n % i = = 0 ):
v.append(i);
# condition to check the
# divisor is not the number itself
if ( int (n / i) ! = i):
v.append( int (n / i))
# return the vector
return v
# Function to check if the number # is abundant or not def checkAbundant(n):
sum = 0
# find the divisors using function
v = factors(n)
# sum all the factors
for i in range ( len (v)):
sum + = v[i]
# check for abundant or not
if ( sum > n):
return True
else :
return False
# Function to check if the # number is semi-perfect or not def checkSemiPerfect(n):
# find the divisors
v = factors(n)
# sorting the vector
v.sort(reverse = False )
r = len (v)
# subset to check if no is semiperfect
subset = [[ 0 for i in range (n + 1 )]
for j in range (r + 1 )]
# initialising 1st column to true
for i in range (r + 1 ):
subset[i][ 0 ] = True
# initialising 1st row except zero position to 0
for i in range ( 1 , n + 1 ):
subset[ 0 ][i] = False
# loop to find whether the number is semiperfect
for i in range ( 1 , r + 1 ):
for j in range ( 1 , n + 1 ):
# calculation to check if the
# number can be made by summation of divisors
if (j < v[i - 1 ]):
subset[i][j] = subset[i - 1 ][j]
else :
subset[i][j] = (subset[i - 1 ][j] or
subset[i - 1 ][j - v[i - 1 ]])
# if not possible to make the
# number by any combination of divisors
if ((subset[r][n]) = = 0 ):
return False
else :
return True
# Function to check for # weird or not def checkweird(n):
if (checkAbundant(n) = = True and
checkSemiPerfect(n) = = False ):
return True
else :
return False
# Driver Code if __name__ = = '__main__' :
n = 70
if (checkweird(n)):
print ( "Weird Number" )
else :
print ( "Not Weird Number" )
# This code is contributed by # Surendra_Gangwar |
// C# program to check if // the number is weird or not using System;
using System.Collections.Generic;
class GFG
{ // code to find all the // factors of the number // excluding the number itself static List< int > factors( int n)
{ // List to store
// the factors
List< int > v = new List< int >();
v.Add(1);
// note that this loop
// runs till sqrt(n)
for ( int i = 2;
i <= Math.Sqrt(n); i++)
{
// if the value of
// i is a factor
if (n % i == 0)
{
v.Add(i);
// condition to check
// the divisor is not
// the number itself
if (n / i != i)
{
v.Add(n / i);
}
}
}
// return the List
return v;
} // Function to check if the // number is abundant or not static Boolean checkAbundant( int n)
{ List< int > v;
int sum = 0;
// find the divisors
// using function
v = factors(n);
// sum all the factors
for ( int i = 0; i < v.Count; i++)
{
sum += v[i];
}
// check for abundant
// or not
if (sum > n)
return true ;
else
return false ;
} // Function to check if the // number is semi-perfect or not static Boolean checkSemiPerfect( int n)
{ List< int > v;
// find the divisors
v = factors(n);
// sorting the List
v.Sort();
int r = v.Count;
// subset to check if
// no is semiperfect
Boolean [,]subset = new Boolean[r + 1,n + 1];
// initialising 1st
// column to true
for ( int i = 0; i <= r; i++)
subset[i,0] = true ;
// initialising 1st row except
// zero position to 0
for ( int i = 1; i <= n; i++)
subset[0,i] = false ;
// loop to find whether
// the number is semiperfect
for ( int i = 1; i <= r; i++)
{
for ( int j = 1; j <= n; j++)
{
// calculation to check
// if the number can be
// made by summation of
// divisors
if (j < v[i-1])
subset[i,j] = subset[i - 1,j];
else {
subset[i,j] = subset[i - 1,j] ||
subset[i - 1,j -
v[i-1]];
}
}
}
// if not possible to make
// the number by any
// combination of divisors
if ((subset[r,n]) == false )
return false ;
else
return true ;
} // Function to check // for weird or not static Boolean checkweird( int n)
{ if (checkAbundant(n) == true &&
checkSemiPerfect(n) == false )
return true ;
else
return false ;
} // Driver Code public static void Main(String []args)
{ int n = 70;
if (checkweird(n))
Console.WriteLine( "Weird Number" );
else
Console.WriteLine( "Not Weird Number" );
} } // This code is contributed by Princi Singh |
<script> // Javascript program to check if the // number is weird or not // code to find all the factors of // the number excluding the number itself function factors(n)
{ // vector to store the factors
var v = [];
v.push(1);
// note that this loop runs till sqrt(n)
for ( var i = 2; i <= Math.sqrt(n); i++) {
// if the value of i is a factor
if (n % i == 0) {
v.push(i);
// condition to check the
// divisor is not the number itself
if (n / i != i) {
v.push(n / i);
}
}
}
// return the vector
return v;
} // Function to check if the number // is abundant or not function checkAbundant(n)
{ var v = [];
var sum = 0;
// find the divisors using function
v = factors(n);
// sum all the factors
for ( var i = 0; i < v.length; i++) {
sum += v[i];
}
// check for abundant or not
if (sum > n)
return true ;
else
return false ;
} // Function to check if the // number is semi-perfect or not function checkSemiPerfect(n)
{ var v = [];
// find the divisors
v = factors(n);
// sorting the vector
v.sort()
var r = v.length;
// subset to check if no is semiperfect
var subset = Array.from(Array(r+1), ()=>Array(n+1));
// initialising 1st column to true
for ( var i = 0; i <= r; i++)
subset[i][0] = true ;
// initialising 1st row except zero position to 0
for ( var i = 1; i <= n; i++)
subset[0][i] = false ;
// loop to find whether the number is semiperfect
for ( var i = 1; i <= r; i++) {
for ( var j = 1; j <= n; j++) {
// calculation to check if the
// number can be made by summation of divisors
if (j < v[i - 1])
subset[i][j] = subset[i - 1][j];
else {
subset[i][j] = subset[i - 1][j] ||
subset[i - 1][j - v[i - 1]];
}
}
}
// if not possible to make the
// number by any combination of divisors
if ((subset[r][n]) == 0)
return false ;
else
return true ;
} // Function to check for // weird or not function checkweird(n)
{ if (checkAbundant(n) == true &&
checkSemiPerfect(n) == false )
return true ;
else
return false ;
} // Driver Code var n = 70;
if (checkweird(n))
document.write( "Weird Number" );
else document.write( "Not Weird Number" );
</script> |
Weird Number
Time Complexity: O(N * number of factors)
Auxiliary Space: O(N * number of factors)
Efficient approach : Space optimization
In previous approach the current value subset[i][j] is only depend upon the current and previous row values of subset matrix. So to optimize the space complexity we use a single 1D array to store the computations.
Implementation:
#include <bits/stdc++.h> using namespace std;
// code to find all the factors of // the number excluding the number itself vector< int > factors( int n)
{ // vector to store the factors
vector< int > v;
v.push_back(1);
// note that this loop runs till sqrt(n)
int sqrtN = sqrt (n);
for ( int i = 2; i <= sqrtN; i++) {
if (n % i == 0) {
v.push_back(i);
if (n / i != i) {
v.push_back(n / i);
}
}
}
// return the vector
return v;
} // Function to check if the number // is abundant or not bool checkAbundant( int n)
{ vector< int > v;
// find the divisors using function
v = factors(n);
int sum = 0;
// sum all the factors
for ( int i = 0; i < v.size(); i++) {
sum += v[i];
}
// check for abundant or not
return sum > n;
} // Function to check if the // number is semi-perfect or not bool checkSemiPerfect( int n)
{ vector< int > v;
// find the divisors
v = factors(n);
sort(v.begin(), v.end());
int r = v.size();
// initialize vector to store
// computations of subproblems
vector< bool > subset(n + 1, false );
// Base Case
subset[0] = true ;
// iterate over subproblems using nested loop
// to get the computaition of current value from
// previous computations
for ( int i = 0; i < r; i++) {
for ( int j = n; j >= v[i]; j--) {
subset[j] = subset[j] || subset[j - v[i]];
}
}
// return answer
return subset[n];
} // Function to check for // weird or not bool checkWeird( int n)
{ return checkAbundant(n) && !checkSemiPerfect(n);
} // Driver code int main()
{ int n = 70;
if (checkWeird(n))
cout << "Weird Number" ;
else
cout << "Not Weird Number" ;
return 0;
} |
import java.util.ArrayList;
import java.util.Collections;
public class Main {
// Function to find all the factors of the number excluding the number itself
static ArrayList<Integer> factors( int n) {
// ArrayList to store the factors
ArrayList<Integer> v = new ArrayList<>();
v.add( 1 );
// Note that this loop runs until sqrt(n)
int sqrtN = ( int ) Math.sqrt(n);
for ( int i = 2 ; i <= sqrtN; i++) {
if (n % i == 0 ) {
v.add(i);
if (n / i != i) {
v.add(n / i);
}
}
}
// Return the ArrayList
return v;
}
// Function to check if the number is abundant or not
static boolean checkAbundant( int n) {
ArrayList<Integer> v;
// Find the divisors using the factors function
v = factors(n);
int sum = 0 ;
// Sum all the factors
for ( int i = 0 ; i < v.size(); i++) {
sum += v.get(i);
}
// Check if it's abundant or not
return sum > n;
}
// Function to check if the number is semi-perfect or not
static boolean checkSemiPerfect( int n) {
ArrayList<Integer> v;
// Find the divisors using the factors function
v = factors(n);
Collections.sort(v);
int r = v.size();
// Initialize an array to store computations of subproblems
boolean [] subset = new boolean [n + 1 ];
// Base Case
subset[ 0 ] = true ;
// Iterate over subproblems using nested loop
// to compute the current value from previous computations
for ( int i = 0 ; i < r; i++) {
for ( int j = n; j >= v.get(i); j--) {
subset[j] = subset[j] || subset[j - v.get(i)];
}
}
// Return the answer
return subset[n];
}
// Function to check if the number is weird or not
static boolean checkWeird( int n) {
return checkAbundant(n) && !checkSemiPerfect(n);
}
// Driver code
public static void main(String[] args) {
int n = 70 ;
if (checkWeird(n))
System.out.println( "Weird Number" );
else
System.out.println( "Not Weird Number" );
}
} |
import math
# Function to find all factors of n excluding itself def factors(n):
v = [ 1 ]
sqrtN = int (math.sqrt(n))
for i in range ( 2 , sqrtN + 1 ):
if n % i = = 0 :
v.append(i)
if n / / i ! = i:
v.append(n / / i)
return v
# Function to check if the number is abundant or not def checkAbundant(n):
v = factors(n)
sum_factors = sum (v)
return sum_factors > n
# Function to check if the number is semi-perfect or not def checkSemiPerfect(n):
v = factors(n)
v.sort()
r = len (v)
subset = [ False ] * (n + 1 )
subset[ 0 ] = True
for i in range (r):
for j in range (n, v[i] - 1 , - 1 ):
subset[j] = subset[j] or subset[j - v[i]]
return subset[n]
# Function to check if the number is weird or not def checkWeird(n):
return checkAbundant(n) and not checkSemiPerfect(n)
# Driver code n = 70
if checkWeird(n):
print ( "Weird Number" )
else :
print ( "Not Weird Number" )
|
using System;
using System.Collections.Generic;
using System.Linq;
class Program {
// Function to find all factors of n excluding itself
static List< int > Factors( int n)
{
List< int > v = new List< int >{ 1 };
int sqrtN = ( int )Math.Sqrt(n);
for ( int i = 2; i <= sqrtN; i++) {
if (n % i == 0) {
v.Add(i);
if (n / i != i) {
v.Add(n / i);
}
}
}
return v;
}
// Function to check if the number is abundant or not
static bool CheckAbundant( int n)
{
List< int > v = Factors(n);
int sumFactors = v.Sum();
return sumFactors > n;
}
// Function to check if the number is semi-perfect or
// not
static bool CheckSemiPerfect( int n)
{
List< int > v = Factors(n);
v.Sort();
int r = v.Count;
bool [] subset = new bool [n + 1];
subset[0] = true ;
for ( int i = 0; i < r; i++) {
for ( int j = n; j >= v[i]; j--) {
subset[j] = subset[j] || subset[j - v[i]];
}
}
return subset[n];
}
// Function to check if the number is weird or not
static bool CheckWeird( int n)
{
return CheckAbundant(n) && !CheckSemiPerfect(n);
}
static void Main()
{
int n = 70;
if (CheckWeird(n)) {
Console.WriteLine( "Weird Number" );
}
else {
Console.WriteLine( "Not Weird Number" );
}
}
} |
function GFG(n) {
const v = [1];
// Note that this loop runs until the square root of n
const sqrtN = Math.sqrt(n);
for (let i = 2; i <= sqrtN; i++) {
if (n % i === 0) {
v.push(i);
if (n / i !== i) {
v.push(n / i);
}
}
}
// Return the array of GFG
return v;
} // Function to check if a number is abundant or not function checkAbundant(n) {
const v = GFG(n);
let sum = 0;
// Sum all the factors
for (let i = 0; i < v.length; i++) {
sum += v[i];
}
// Check if the number is abundant
return sum > n;
} // Function to check if a number is // semi-perfect or not function checkSemiPerfect(n) {
const v = GFG(n).sort();
const r = v.length;
// Initialize an array to store
// computations of subproblems
const subset = new Array(n + 1).fill( false );
// Base Case
subset[0] = true ;
for (let i = 0; i < r; i++) {
for (let j = n; j >= v[i]; j--) {
subset[j] = subset[j] || subset[j - v[i]];
}
}
// Return the result
return subset[n];
} // Function to check if a number is weird or not function checkWeird(n) {
return checkAbundant(n) && !checkSemiPerfect(n);
} // Driver code function main() {
const n = 70;
if (checkWeird(n)) {
console.log( "Weird Number" );
} else {
console.log( "Not Weird Number" );
}
} // Call the main function main(); |
Weird Number
Time Complexity: O(N * number of factors)
Auxiliary Space: O(N)