Weekly #3- Amicable Numbers: Communicating Math

In order to understand what an amicable number is you must first know what a proper factor is. A proper factor is the sum of the factors of a number, not including that number. An amicable number is when the sum of the proper factors of two numbers are each other. The first pair of amicable numbers is: 220 and 284.

Example: Take 220 and 284 and find the factors of each number.

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

284: 1, 2, 4, 71, 142, 284

You can see that the sum of the factors for each number add up to 504. 
Next, find the proper factors of each number.

220: 1+2+4+5+10+11+20+44+55+110=284

284: 1+2+4+71+142=220

We know by definition, that 220 and 284 are amicable numbers, therefore an amicable pair.

Thabit ibn Qurra's Formula: T(n)=3 * 2^n -1 other wise known as a 321 number.
n=2, 4, and 7.
2^n * T(n) * T(n-1) and 2^n * (9*2^(2n-1) -1) are amicable


Amicable pairs interesting facts:

• For some amicable pairs the sum of their digits added up are equal. Ex: 69615 and 87633. 6+9+6+1+5=27 and 8+7+6+3+3=27
• Harshad numbers are numbers which are divisible by their sum of the digits.
  Harshad Amicable pairs, for some amicable pairs where both numbers are divisible by their sums. Ex: 2620 and 2924. 2+6+2+0=10 2620/10=262 and 2+9+2+4=17 2914/17=172

References:
http://www.shyamsundergupta.com/amicable.htm
http://mathworld.wolfram.com/AmicablePair.html