## Wednesday, April 9, 2008

### Project Euler Problem 21

Ok, this is it- problem description:
 Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.Evaluate the sum of all the amicable numbers under 10000.

Fun part,- C# solution:

using System;class Program{    static long DivSum(long n)    {        long un = 1 + (long)Math.Sqrt(n);        long sum = 1;        for (int i = 2; i <= un; i++)        {            if (n % i == 0)                sum += i + (n / i);        } return sum;    }    static long AmicablePair(long n)    {        long ds = DivSum(n);        long os = DivSum(ds);        long ret;        ret = (os == n && n != ds) ? ds : 0;        return ret;    }    static long AmicableNumbersSum(long until)    {        bool[] amic = new bool[until];        long nextAmic;        long sum = 0;        for (int i = 1; i < until; i++)        {            if (!amic[i])            {                nextAmic = AmicablePair(i);                if (nextAmic > 0)                {                    if (nextAmic < until) amic[nextAmic] = true;                    sum += i + nextAmic;                }            }        }        return sum;    }    static void Main(string[] args)    {        System.Diagnostics.Stopwatch sw = new System.Diagnostics.Stopwatch();        sw.Start();        Console.WriteLine("Amicable numbers sum {0}", AmicableNumbersSum(10000));        sw.Stop();        Console.WriteLine("Calculation time {0} ms", sw.ElapsedMilliseconds);        Console.ReadLine();    } }

So solution is fast enough,- runs in about 20 ms. A little bit explanations of why this
solution is fast:

• We need not to search all divisors, but rather until SQRT limit of number.
This is because if number, lets say x, have divisor d1 above SQRT(x), then,
that number x MUST have also a divisor d2 = x/d1, below SQRT(x) limit.
• We need not to verify are all test numbers amicable, because if we find one
amicable number, lets say x, then we know that there exists second amicable
number y, which is sum of proper divisors of x. This rule is true, because
we acctually searching amicable number pairs. So by applying this rule
we can use memorization - mark second amicable number in bool array,
and skip that number from test later.

Have fun!

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