Frugal Number in Java

In this section, we will learn what is a frugal number and also create Java programs to check if the given number is a frugal number or not. The frugal number program is frequently asked in Java coding interviews and academics.

Frugal Number

A frugal number is a natural number n with base-b whose number of digits is greater than the number of digits in its prime factorization (including exponents greater than 1). It is an OEIS sequence A046759.

Points to remember

The union of frugal and equidigital numbers gives results in economical
Some scholar calls the frugal numbers economical.
Number greater than 1923 can be written as a sum of frugal numbers.

We can say that frugal number is the same as the economical number. Hence, we can represent frugal numbers as write n as a product of primes raised to powers.

Where:

d(n): Represents the number of digits in the product.

l(n): Represents the number of digits in n.

Therefore, sequence gives n such that d(n)<l(n).

Let's understand it through the examples.

Frugal Number Example

125 = 5³, l(n) = 3 and d(n) = 2. Here, d(n)<l(n), so 125 is a frugal number.

128 = 2⁷, l(n) = 3 and d(n) = 2. Here, d(n)<l(n), so 128 is a frugal number.

243 = 3⁵, l(n) = 3 and d(n) = 2. Here, d(n)<l(n), so 243 is a frugal number.

256 = 2⁸, l(n) = 3 and d(n) = 2. Here, d(n)<l(n), so 256 is a frugal number.

2048 = 2¹¹, l(n) = 4 and d(n) = 3. Here, d(n)<l(n), so 2048 is a frugal number.

We observe that the number written with k digits, while its factorization of the form p^q uses digits greater than k.

Similarly, we can check other numbers also.

Some other frugal numbers are in base 10 are 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2560, 2809, 3125, 3481, 3584, 3645, 3721, 4096, 4374, 4375, 4489, 4802, 4913.

The first frugal number that is not a prime power is 1029 = 3 × 7³.

Frugal Number Java Program

FrugalNumberExample.java

import java.io.*;
import java.util.*;
public class FrugalNumberExample
{
//function checks if the number is frugal or not
static boolean frugal(long n)
{
ArrayList<Long> arr = primes(n);
long t = n;
//determines number of digits in prime factorization of the number
long s = 0;
for (int i = 0; i < arr.size(); i++)
{
if (t % arr.get(i) == 0)
{
//exponent for current factor
long k = 0;
//loop counts the number of times the prime factor divides
//i.e. determines exponent
while (t % arr.get(i) == 0)
{
t = t / arr.get(i);
k++;
}
//determines the number of digits in the exponent avoiding exponents of value 1
if (k == 1)
s = s + countDigits(arr.get(i));
else if (k != 1)
s = s + countDigits(arr.get(i)) + countDigits(k);		
}
}
//the given number will be frugal number if the condition returns true
return (countDigits(n) > s && s != 0);
}    
//function that finds primes up to given numbers
static ArrayList<Long> primes(long n)
{
//creates a boolean array for primes    
boolean []prime = new boolean[(int)n + 1];
for(int i = 0; i < n + 1; i++)
prime[i] = true;
//determines prime by using the Sieve of Eratosthenes method
for (int i = 2; i * i <= n; i++)
{
//if prime[i] is true then it is prime
if (prime[i] == true)
{
//finds multiples of p
for (int j = i * 2; j <= n; j += i)
prime[j] = false;
}
}
//creates an array of prime numbers
ArrayList<Long> array = new ArrayList<Long>();
for (int i = 2; i < n; i++)
//returns true if i is prime
if (prime[i])
//adding elements to the array
array.add((long)i);
//returns an array of prime numbers
return array;
}
//function counts the number of digits in n
static int countDigits(long n)
{
long temp = n;
int count = 0;
while (temp != 0)
{
//divides the number by 10    
temp = temp / 10;
//increments the variable c by 1
count++;
}
//returns the number of digits
return count;
}
//driver code
public static void main(String args[])
{
Scanner sc = new Scanner(System.in);
System.out.print("Enter the number to check: ");
long num = sc.nextLong();
//function calling
if (frugal(num))
System.out.println("It is a frugal number.");
else
System.out.println("It is not a frugal number.");
}
}

Output 1:

Enter the number to check: 125
It is a frugal number.

Output 2:

Enter the number to check: 2084
It is not a frugal number.

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