Joel Bradbury
wrote this thing

Friendly Numbers

August 24th, 2010 4 years ago

This is in reponse to this xkcd comic.

So what are friendly numbers?

We need first to get define a divisor function over the integers, written σ(n) if you’re so inclined. To get it first we get all the integers that divide into n. So for 3, it’s 1 and 3. For 4, it’s 1, 2, and 4, and for 5 it’s only 1 and 5.

Now sum them to get σ(n). So σ(3) = 1 + 3 = 4, or σ(4) = 1 + 2 + 4 = 6, and so on.

For each of these n, there is something called a characteristic ratio. Now that’s just the divors function over the integer itself ( σ(n)/n . So the characteristic ratio where n = 6 is σ(6)/6 = 12/6 =2.

Once you have the characteristic ratio for any integer n, any other integers that share the same chacteristic are called friendly with each other. So to put it simply a friendly number is any integer that shares its characteristic ratio with at least one other integer. The converse of that is called a solitary number, where it doesn’t share it’s characteristic with anyone else.

1,2,3,4 and 5 are solitary. 6 is friendly with 28. ( σ(6)/6 = (1+2+3+6)/6 = 12/6 = 2 = 56/28 = (1+2+4+7+14+28)/28 = σ(28)/28.

You may be now be thinking the following:

  1. What are they good for?
  2. Who cares?

So what are they good for?

Not much. But it’s interesting. While on the face of it, it all seems very straight forward it quickly becomes far more complex than you would think.

Some numbers are easily proved to be solitary. Other’s we know to be friendly. But there are many more we don’t know. Seriously look at the following list of friendly numbers :

6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

That list isn’t an abridged list. That’s the complete list of friendly numbers humanity can prove. What about numbers 10, 14, 15 or 20? Can’t say. There are a number of theorms for predicting where solitary numbers will occur, but as yet nothing’s been proven.

Have a go. It’s far harder than it looks.

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