28

A "sociable chain" is a set of numbers where each number is the sum of the proper factors of the previous number, and then it gets back to the beginning number.


For example: 

For example
* The proper factors of 12496 are:  1, 2, 4, 8, 11, 16, 22, 44, 71, 88, 142, 176, 284, 568, 781, 1136, 1562, 3124, 6248.  If you add them up, you get 14288,
* The proper factors of 14288 add up to 15472.
* The proper factors of 15472 add up to 14536.
* The proper factors of 14536 add up to 14264.
* The proper factors of 14264 add up to 12496.  Which is where we started.  So this is a sociable chain of 5 numbers.

When the chain ony has 2 numbers, it is an  "amicable pair".
* The proper factors of 12285 are 1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095.  These add up to 14595.

* The proper factors of 14595 add up to 12285.

SO, what does this have to do with 28?  The longest known chain has 28 numbers in it:

14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946
22976, 22744, 19916, 17716, back to 14316 which I don’t count a second time.