In this section, we will explore the concept of Smith numbers and develop Java programs to determine whether a given number is a Smith number. The topic of Smith numbers often appears in Java coding tests and academic assignments. Our focus will be on understanding the characteristics of Smith numbers and implementing code to check if a given number qualifies as a Smith number.
What is a Smith Number?
A Smith number, also known as a joke number, is a composite number that possesses the unique property of having the sum of its digits equal to the sum of the digits of its prime factors, excluding the number 1. Smith numbers can be found in the sequence OEIS A006753. In our exploration, we will delve into the characteristics of Smith numbers and develop Java programs to determine whether a given number qualifies as a Smith number.
Sum of digits of n = Sum of digits of prime factors of n (counted with multiplicity)
Smith Number Example
Example 1: Suppose, we want to check whether the number 85 is smith or not.
The sum of digits = 8 + 5 = 13
Prime factors of 85 are 5,17
The sum of digits of its prime factors = 5 + 1+ 7 = 13
Compare the sum of digits with the sum of digits of its prime factors i.e. 13=13. Both are equal. Hence, the given number (85) is a Smith number.
Example 2: Let’s check whether another number 999 is Smith or not.
The Sum of digits = 9+ 9+9 = 27
Prime factors of 999 are: 3×3×3,37
The sum of digits of its prime factors = 3+3+3+3+7 =19
Compare the sum of digits with the sum of digits of its prime factors i.e. 27≠19. Hence, the given number (999) is not a Smith number.
Steps to Find Smith Number
- Read or assign a value to the variable representing the number from the user.
- Calculate the sum of the digits of the given number.
- Find the prime factors of the given number.
- Determine the sum of the digits of the prime factors.
- Compare the sum of the digits of the given number with the sum of the digits of its prime factors.
- If the sums are equal, then the given number is a Smith number.
- Otherwise, it is not a Smith number.
Smith Number Program in Java
SmithNumberExample.java
import java.util.*; public class SmithNumberExample { //function finds the sum of digits of its prime factors static int findSumPrimeFactors(int n) { int i=2, sum=0; while(n>1) { if(n%i==0) { sum=sum+findSumOfDigit(i); n=n/i; } else { do { i++; } while(!isPrime(i)); } } //returns the sum of digits of prime factors return sum; } //function finds the sum of digits of the given numbers static int findSumOfDigit(int n) { //stores the sum int s=0; while(n>0) { //finds the last digit of the number and add it to the variable s s=s+n%10; //removes the last digit from the given number n=n/10; } //returns the sum of digits of the number return s; } //function checks if the factor is prime or not static boolean isPrime(int k) { boolean b=true; int d=2; while(d<Math.sqrt(k)) { if(k%d==0) { b=false; } d++; } return b; } //driver code public static void main(String args[]) { Scanner sc = new Scanner(System.in); System.out.print("Enter a number: "); //reads an integer from the user int n=sc.nextInt(); //calling the user-defined function that finds the sum of digits of the given number int a = findSumOfDigit(n); //calling the user-defined function that finds the sum of prime factors int b = findSumPrimeFactors(n); System.out.println("Sum of Digits of the given number is = "+a); System.out.println("Sum of digits of its prime factors is = "+b); //compare both the sums if(a==b) //prints if above condition returns true System.out.print("The given number is a smith number."); //prints if above condition returns false else System.out.print("The given number is not a smith number."); } }
Output 1:
Enter a number: 265 Sum of Digits of the given number is = 13 Sum of digits of its prime factors is = 13 The given number is a smith number.
Output 2:
Enter a number: 2805 Sum of Digits of the given number is = 15 Sum of digits of its prime factors is = 18 The given number is not a smith number.
SmithNumberExample1.java
import java.util.*; public class SmithNumberExample1 { //function finds all the prime factors for the given number static List<Integer> findPrimeFactors(int n) { //creating an array list that stores the prime factors List<Integer> result = new ArrayList<>(); for (int i = 2; n % i == 0; n = n/i) result.add(i); for (int i = 3; i * i <= n; i =i+2) { while (n % i == 0) { result.add(i); n = n/i; } } if (n != 1) //the add() method adds the prime factors to the list result.add(n); return result; } //function finds the sum of digits of the given number static int sumOfDigits(int n) { int sum = 0; while (n > 0) { //finds the last digit and add it to the sum sum =sum+(n % 10); //removes the last digit from the number n = n/10; } //returns the sum of digits return sum; } //driver code public static void main(String args[]) { //finds all smit numbers up to 10000 for (int n = 1; n < 10000; n++) { //calling the user-defined function that finds prime factors List<Integer> factors = findPrimeFactors(n); //the size() method returns the number of elements in the list //executes until the condition becomes false if (factors.size() > 1) { int sum = sumOfDigits(n); for (int f : factors) sum =sum-sumOfDigits(f); if (sum == 0) System.out.println(n); } } } }
Output:
4 22 27 58 85 94 121 166 202 265 274 319 346 355 378 382 391 438 454 483 517 526 535 562 576 588 627 634 636 645 648 654 663 666 690 706 728 729 762 778 825 852 861 895 913 915 922 958 985 1086 1111 1165 1219 1255 1282 1284 1376 1449 1507 1581 1626 1633 1642 1678 1736 1755 1776 1795 1822 1842 1858 1872 1881 1894 1903 1908 1921 1935 1952 1962 1966 2038 2067 2079 2155 2173 2182 2218 2227 2265 2286 2326 2362 2366 2373 2409 2434 2461 2475 2484 2515 2556 2576 2578 2583 2605 2614 2679 2688 2722 2745 2751 2785 2839 2888 2902 2911 2934 2944 2958 2964 2965 2970 2974 3046 3091 3138 3168 3174 3226 3246 3258 3294 3345 3366 3390 3442 3505 3564 3595 3615 3622 3649 3663 3690 3694 3802 3852 3864 3865 3930 3946 3973 4054 4126 4162 4173 4185 4189 4191 4198 4209 4279 4306 4369 4414 4428 4464 4472 4557 4592 4594 4702 4743 4765 4788 4794 4832 4855 4880 4918 4954 4959 4960 4974 4981 5062 5071 5088 5098 5172 5242 5248 5253 5269 5298 5305 5386 5388 5397 5422 5458 5485 5526 5539 5602 5638 5642 5674 5772 5818 5854 5874 5915 5926 5935 5936 5946 5998 6036 6054 6084 6096 6115 6171 6178 6187 6188 6252 6259 6295 6315 6344 6385 6439 6457 6502 6531 6567 6583 6585 6603 6684 6693 6702 6718 6760 6816 6835 6855 6880 6934 6981 7026 7051 7062 7068 7078 7089 7119 7136 7186 7195 7227 7249 7287 7339 7402 7438 7447 7465 7503 7627 7674 7683 7695 7712 7726 7762 7764 7782 7784 7809 7824 7834 7915 7952 7978 8005 8014 8023 8073 8077 8095 8149 8154 8158 8185 8196 8253 8257 8277 8307 8347 8372 8412 8421 8466 8518 8545 8568 8628 8653 8680 8736 8754 8766 8790 8792 8851 8864 8874 8883 8901 8914 9015 9031 9036 9094 9166 9184 9193 9229 9274 9276 9285 9294 9296 9301 9330 9346 9355 9382 9386 9387 9396 9414 9427 9483 9522 9535 9571 9598 9633 9634 9639 9648 9657 9684 9708 9717 9735 9742 9760 9778 9840 9843 9849 9861 9880 9895 9924 9942 9968 9975 9985
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