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Calculate amicable numbers efficiently to a very high upper limit in java

I have a program that computes amicable pairs of numbers up to a certain limit and prints them out to Stdout. FYI - two numbers are amicable if each is equal to the sum of the proper divisors of the other (for example, 220 and 284).
My program works fine for the first 8 or so pairs, but the problem is when the limit gets to a very high number e.g 5 million it becomes very slow. Is there a way to optimize this so that it computes much faster?
Here is my code
public class Amicable{
private static int num1, num2, limit = 5000000;
public static void main(String[] args){
for(int i = 1; i < limit;i++){
num1= sumOfDivisors(i);
num2 = sumOfDivisors(i)
if(num1 == num2){
System.out.println(num2 + " " + num1);
}
}
}
}
private static int sumOfDivisors(int n){
int sum = 0;
for(int i=1;i<=n/2;i++){
if(n%i==0){
sum =sum+i;
}
}
return sum;
}
}

You should make the innermost loop efficient.
static int sumOfDivisors(int n) {
int sum = 1;
int max = (int) Math.sqrt(n);
if (n > 1 && max * max == n) // in case of n is perfect square number
sum -= max;
for (int i = 2; i <= max; ++i)
if (n % i == 0)
sum += i + n / i;
return sum;
}
If n = 220, it is divisible by 2, so you can add 2 and 110 to the sum at the same time. However, if n is a perfect square number, for example n = 100, it is divisible by 10, but if 10 and 10 are added, they will duplicate, so subtract it in advance.
output for limit =2000000:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416
63020 76084
66928 66992
.....
1468324 1749212
1511930 1598470
1669910 2062570
1798875 1870245
It took 7 seconds on my PC.

You could change the statement of "if(i<num1)" to "if(i<num1 || num1<limit)".

This is a more efficient and faster approach.
Inside sumOfDivisors:
Start iteration from 2. (no a big deal) and add 1 during return time with sum as 1 is also a divisor.
Modify loop termination logic, i <= Math.sqrt(n).
For any number ‘num’ all its divisors are always less than and equal to ‘num/2’ and all prime factors are always less than and equal to sqrt(num).
If n%i == 0, then check, both divisors are are equal or not. If equal, take one if not equal then take both one.
private static int sumOfDivisors(int n) {
int sum = 0;
int sqrt = Math.sqrt(n);
for (int i = 2; i <= sqrt; i++) {
if (n % i == 0) {
if(i == (n/i))
sum = sum + i;
else
sum += (i+ n/i);
}
}
return sum+1;
}

Java:Three digit Sum - Find out all the numbers between 1 and 999 where the sum of 1st digit and 2nd digit is equal to 3rd digit

Problem statement: Three digit sum - Find all the numbers between 1 and 999 where the sum of the 1st digit and the 2nd digit is equal to the 3rd digit.
Examples:
123 : 1+2 = 3
246 : 2+4 = 6
Java:
public class AssignmentFive {
public static void main(String[] args) {
int i=1;
int valuetwo;
int n=1;
int sum = 0;
int valuethree;
int valueone = 0;
String Numbers = "";
for (i = 1; i <= 999; i++) {
n = i;
while (n > 1) {
valueone = n % 10;/*To get the ones place digit*/
n = n / 10;
valuetwo = n % 10;/*To get the tens place digit*/
n = n / 10;
valuethree = n;/*To get the hundreds place digit*/
sum = valuethree + valuetwo;/*adding the hundreds place and
tens place*/
}
/*Checking if the ones place digit is equal to the sum and then print
the values in a string format*/
if (sum == valueone) {
Numbers = Numbers + n + " ";
System.out.println(Numbers);
}
}
}
}
I got my result :
1
10
100
1000
10000
100000
1000000
10000000
100000000
1000000000
10000000001
100000000011
1000000000111
10000000001111
100000000011111
1000000000111111
10000000001111111
100000000011111111
1000000000111111111
Process finished with exit code 0
The result is not showing the actual result like it should be which should show values like: 123, 246 (Please refer to the problem statement above.)
Please let me know what seems to be the issue with the code and how to tweak it.

Don't know what you're trying to do with that while loop, or why you are building up a space-separated string of numbers.
Your code should be something like:
for (int n = 1; n <= 999; n++) {
int digit1 = // for you to write code here
int digit2 = // for you to write code here
int digit3 = // for you to write code here
if (digit1 + digit2 == digit3) {
// print n here
}
}

So basically your question is how to calculate the numbers, right?
My first hint for you would be how to get the first, second and third value from a 2 or 3 digit number.
For example for 3 digits you can do int hundretDigit = (n - (n % 100)) % 100. Of course this is really inefficient. But just get code working before optimizing it ;)
Just think about a way to get the "ten-digit" (2nd number). Then you add them and if they equal the third one you write System.out.println(<number>);
EDIT:
For 2 digit numbers I will give you the code:
if(i >= 10 && i <= 99) {
int leftDigit = (i - (i % 10)) / 10;
if(leftDigit == (i % 10)) {
//Left digit equals right digit (for example 33 => 3 = 3
System.out.println(i);
}
}
Try again and edit your source code. If you have more questions I will edit my (this) answer to give you a little bit more help if you need!

What would be the solution for this question if it had strong test cases?

https://www.codechef.com/problems/PRIME1
If you don't wish to open the link here's a short description of the question below :
This problem asks us to print all prime number within a given range.
There are 10 test cases and each one will provide us a start and end value of a range.
The start and end of this range can take values between 1 and 10^9.
The difference between the start and end values is 10^5 or lesser.
The time limit for the problem is 2 seconds. (that is, for all 10 test cases together)
My thinking on this:
A common estimate is that the online judge used by Codechef can perform ~10^7 operations in 1 second.
We have 10 test cases and in the worst case each one will have a range of 10^5 (since that's the max range given). Now,
10*(10^5)= 10^6 , which is the max number of operations we can perform in 1 second, so for each number in the range we must identify if it is prime in O(1).
Approaches:
1. Simple method for testing primality - Iterate through all numbers from 2 to n-1 and for every number check if it divides n
Ans: Won't work because for the worst case,
= (numbers of the highest size) * (total numbers in max range) * (total test cases)
= (10^9 * 10^5) * 10
= 10^15
2. Square root method to check if prime
Ans: Won't work because, in the worst case,
= (calculating sq. root of numbers of size 10^9) * (total numbers in max range) * (total test cases)
= (~10^4) * (10^5) * 10
= 10^10
3. Using Sieve of Eratosthenes
Precompute primes from 1 to 32000 (this number because it is approx the sq. root of 10^9)
Then to check of a value within the range is primeor not-
if value is between 1 and 32000
directly refer the precomputed value
else
try dividing that value by all precomputed primes, if it divides evenly then its not a prime
Ans: won't work because, in the worst case,
= (number of primes between 1 and 32000) *(total numbers in max range) * (total test cases)
= (3234) * (10^5) * (10)
= 10^9
Code for approach 3:
import java.util.*;
import java.io.*;
class Main
{
static ArrayList<Integer> sieve(ArrayList<Integer> primes)
{
int[] prime=new int[32001];
for(int i=2; i<32001; i++)
{
if(prime[i]==0)
{
for(int j=i+i; j<32001; j+=i)
{
prime[j]=1;
}
}
}
for(int i=2; i<32001; i++)
{
if(prime[i]==0)
{
primes.add(i);
}
}
return primes;
}
public static void main(String[] args)
{
int t,m,n,flag;
ArrayList<Integer> primes= new ArrayList<Integer>();
FastReader scanner= new FastReader();
t=scanner.nextInt();
primes= sieve(primes);
while(t-- > 0)
{
m=scanner.nextInt();
n=scanner.nextInt();
for(int i=m; i<=n; i++)
{
if(i < 32001)
{
if(primes.contains(i))
{
System.out.println(i);
}
}
else
{
flag=0;
for(int j=0; j<primes.size(); j++)
{
if(i%primes.get(j) == 0)
{
flag=1;
break;
}
}
if(flag==0)
{
System.out.println(i);
}
}
}
System.out.println();
}
}
}
While approach 1 obviously didn't work, approach 2 and 3 surprisingly passed!
I'm guessing it passed because the test cases for the problem were weak.
A strong test case would be something like:
10
999900000 1000000000
999899999 999999999
999899998 999999998
999899997 999999997
999899996 999999996
999899995 999999995
999899994 999999994
999899993 999999993
999899992 999999992
999899991 999999991
I ran approach 3 for this test case and it is always taking more than 2 seconds to compute.
If this question did have strong test cases, what would be the correct approach to solve it with the given constraints?

If you going to try figure out prime numbers in a range use the Sieve of Eratosthene algorithm. The basic premise of Sieve is that give a range of numbers you eliminate all numbers that are multiples of prime factors (i.e once we establish that 2 is prime, we eliminate all its multiples ...4, 6, 8, etc)
A implementation of this would be as follows:
private void printPrimes(int max) {
int [] prime = new int[max + 1];
for (int i = 1; i <= max; i++) {
prime[i] = i; // Assume everything is prime initially
}
// if number is prime, then multiples of that factor are not prime
for (int f = 2; f * f <= max; f++) {
if (prime[f] != 0) {
for (int j = f; f * j <= max; j++) {
prime[f * j] = 0;
}
}
}
int counter = 0;
for (int i = 1; i <= max; i++) {
if (prime[i] != 0) counter++;
}
prime = Arrays.stream(prime).filter(i -> i != 0).toArray();
System.out.println("There are " + counter + " primes between 1 and " + max);
System.out.println(Arrays.toString(prime));
}

In Java how to find random factors of a given number

I have a given number. How can I find the factors of that number (for example, 5 and 3 for the number 15)? Here is the code I tried:
int factor1 = 2;
while ((a % factor1 != 0) && (a >= factor1)) {
d++;
}
if (factor1 == a){
d = 1;
}
But this gives me only the smallest factor (i.e a=3 all the time). I would like to get a random set of factors.

Loop through each number from 1 to N inclusively using the modulus operator (%). If n%currentNumber==0, they are a factor. Below, I did this using a for loop, outputting each factor as it is found.
int number=15;
for(int i = 1; i <= number; i++){
if(number%i==0){
System.out.println("Found factor: " + i);
}
}
As Theo said in a comment on this post, you can also use number/2, and arbitrarily include 1 and number.
int number=2229348;
System.out.println("Found factor: " + 1);
for(int i = 2; i <= number/2; i++){
if(number%i==0){
System.out.println("Found factor: " + i);
}
}
System.out.println("Found factor: " + number);

You can iterate through the numbers from 2 to a/2 and check if the given number divides a, which is done using the % operator:
int a = 15;
System.out.print("Divisors of " + a + ": ");
for(int i = 2; i <= a/2; ++i) {
if(a % i == 0) {
System.out.print(i + " ");
}
}
System.out.println();
This code prints all of the divisors of a. Not that you most probably want to ignore 1, since it divides all integers. Moreover, you don't need to check the numbers until a, because no number bigger than a / 2 can actually divide a apart from a itself.

The while loop with default values of a=15 and multiple=2 is already in an infinite loop. You need to correct that and check for subsequent increments of multiple whenever a%multiple ! = 0
public class Factors {
public static void main(String[] args){
/**
int multiple1=2,d=0,a=15; //multiple to be rephrased as factor
while((a%multiple1 != 0)&&(a>=multiple1)){
multiple1++; //this will increment the factor and check till the number itself
//System.out.println(multiple1+" is not a factor of a");
}
if(multiple1==a){
d=1;
}
*commented the original code
*/
int factor=1,d=0,a=20; //multiple rephrased as factor
while(a/2>=factor){ //re-arranged your while condition
factor++;
if((a%factor==0))
d++; //increment factor count whenever a factor is found
}
System.out.println("Total number of factors of a : "+(d+2)); // 1 and 'a' are by default factors of number 'a'
}
}

To find all factors inlcuding 1 and the number itself you can do something like below:
//Iterate from 2 until n/2 (inclusive) and divide n by each number.
//Return numbers that are factors (i.e. remainder = 0). Add the number itself in the end.
int[] findAllFactors(int number) {
int[] factors = IntStream.range(1, 1 + number / 2).filter(factor -> number % factor == 0).toArray();
int[] allFactors = new int[factors.length+1];
System.arraycopy(factors,0,allFactors,0,factors.length);
allFactors[factors.length] = number;
return allFactors;
}
To find only prime factors you can do something like this:
//Iterate from 2 until n/2 (inclusive) and divide n by each number.
// Return numbers that are factors (i.e. remainder = 0) and are prime
int[] findPrimeFactors(int number) {
return IntStream.range(2, 1 + number/ 2).filter(factor -> number % factor == 0 && isPrime(factor)).toArray();
}
Helper method for primality check:
//Iterate from 2 until n/2 (inclusive) and divide n by each number. Return false if at least one divisor is found
boolean isPrime(int n) {
if (n <= 1) throw new RuntimeException("Invalid input");
return !IntStream.range(2, 1+n/2).filter(x -> ((n % x == 0) && (x != n))).findFirst().isPresent();
}
If you are not on Java 8 and/or not using Lambda expressions, a simple iterative loop can be as below:
//Find all factors of a number
public Set<Integer> findFactors(int number) {
Set<Integer> factors = new TreeSet<>();
int i = 1;
factors.add(i);
while (i++ < 1 + number / 2) {
if ((number % i) == 0) {
factors.add(i);
}
}
factors.add(number);
return factors;
}

public class Abc{
public static void main(String...args){
if(args.length<2){
System.out.println("Usage : java Abc 22 3");
System.exit(1);
}
int no1=Integer.parseInt(args[0]);
int no=Integer.parseInt(args[1]),temp=0,i;
for(i=no;i<=no1;i+=no){
temp++;
}
System.out.println("Multiple of "+no+" to get "+no1+" is :--> "+temp);
//System.out.println(i+"--"+no1+"---"+no);
System.out.println("Reminder is "+(no1-i+no));
}
}

How does this prime number test in Java work?

The code snippet below checks whether a given number is a prime number. Can someone explain to me why this works? This code was on a study guide given to us for a Java exam.
public static void main(String[] args)
{
int j = 2;
int result = 0;
int number = 0;
Scanner reader = new Scanner(System.in);
System.out.println("Please enter a number: ");
number = reader.nextInt();
while (j <= number / 2)
{
if (number % j == 0)
{
result = 1;
}
j++;
}
if (result == 1)
{
System.out.println("Number: " + number + " is Not Prime.");
}
else
{
System.out.println("Number: " + number + " is Prime. ");
}
}

Overall theory
The condition if (number % j == 0) asks if number is exactly divisible by j
The definition of a prime is
a number divisible by only itself and 1
so if you test all numbers between 2 and number, and none of them are exactly divisible then it is a prime, otherwise it is not.
Of course you don't actually have to go all way to the number, because number cannot be exactly divisible by anything above half number.
Specific sections
While loop
This section runs through values of increasing j, if we pretend that number = 12 then it will run through j = 2,3,4,5,6
int j = 2;
.....
while (j <= number / 2)
{
........
j++;
}
If statement
This section sets result to 1, if at any point number is exactly divisible by j. result is never reset to 0 once it has been set to 1.
......
if (number % j == 0)
{
result = 1;
}
.....
Further improvements
Of course you can improve that even more because you actually need go no higher than sqrt(number) but this snippet has decided not to do that; the reason you need go no higher is because if (for example) 40 is exactly divisible by 4 it is 4*10, you don't need to test for both 4 and 10. And of those pairs one will always be below sqrt(number).
It's also worth noting that they appear to have intended to use result as a boolean, but actually used integers 0 and 1 to represent true and false instead. This is not good practice.

I've tried to comment each line to explain the processes going on, hope it helps!
int j = 2; //variable
int result = 0; //variable
int number = 0; //variable
Scanner reader = new Scanner(System.in); //Scanner object
System.out.println("Please enter a number: "); //Instruction
number = reader.nextInt(); //Get the number entered
while (j <= number / 2) //start loop, during loop j will become each number between 2 and
{ //the entered number divided by 2
if (number % j == 0) //If their is no remainder from your number divided by j...
{
result = 1; //Then result is set to 1 as the number divides equally by another number, hergo
} //it is not a prime number
j++; //Increment j to the next number to test against the number you entered
}
if (result == 1) //check the result from the loop
{
System.out.println("Number: " + number + " is Not Prime."); //If result 1 then a prime
}
else
{
System.out.println("Number: " + number + " is Prime. "); //If result is not 1 it's not a prime
}

It works by iterating over all number between 2 and half of the number entered (since any number greater than the input/2 (but less than the input) would yield a fraction). If the number input divided by j yields a 0 remainder (if (number % j == 0)) then the number input is divisible by a number other than 1 or itself. In this case result is set to 1 and the number is not a prime number.

Java java.math.BigInteger class contains a method isProbablePrime(int certainty) to check the primality of a number.
isProbablePrime(int certainty): A method in BigInteger class to check if a given number is prime.
For certainty = 1, it return true if BigInteger is prime and false if BigInteger is composite.
Miller–Rabin primality algorithm is used to check primality in this method.
import java.math.BigInteger;
public class TestPrime {
public static void main(String[] args) {
int number = 83;
boolean isPrime = testPrime(number);
System.out.println(number + " is prime : " + isPrime);
}
/**
* method to test primality
* #param number
* #return boolean
*/
private static boolean testPrime(int number) {
BigInteger bValue = BigInteger.valueOf(number);
/**
* isProbablePrime method used to check primality.
* */
boolean result = bValue.isProbablePrime(1);
return result;
}
}
Output: 83 is prime : true
For more information, see my blog.

Do try
public class PalindromePrime {
private static int g ,k ,n =0,i,m ;
static String b ="";
private static Scanner scanner = new Scanner( System.in );
public static void main(String [] args) throws IOException {
System.out.print(" Please Inter Data : ");
g = scanner.nextInt();
System.out.print(" Please Inter Data 2 : ");
m = scanner.nextInt();
count(g,m);
}
//
//********************************************************************************
private static int count(int L, int R)
for( i= L ; i<= R ;i++){
int count = 0 ;
for( n = i ; n >=1 ;n -- ){
if(i%n==0){
count = count + 1 ;
}
}
if(count == 2)
{
b = b +i + "" ;
}
}
System.out.print(" Data : ");
System.out.print(" Data : \n " +b );
return R;
}
}