friendly numbers

friendly numbers

This is in reponse to this comic.

So what are friendly numbers? Lets do this quick.

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. That’s all well and good, but what are they good for?
  2. Who cares?

So what are they good for?

I dunno. But it’s interesting no? Actually while on the face of it, it all seems quite 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 my-o-my there are alot more we don’t know. I mean look. Here is the complete list of friendly numbers we know about. And I mean COMPLETE.

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’s it. What about 10, 14, 15, or 20? Don’t know. Infact it’s on the list of unsolved Mathematical problems at wolfram. Doubt there’s a cash prize involved but why not give it a go. It’s harder that it looks.

Wow.
Well done. You read all the way to the end. To give you a treat I present this lovely video. It’s a visualization of Lovely Head by Goldfrapp. Yes it’s got lot’s of maths in. No it doesn’t have anything to do with friendly numbers.

Or even better, watch it in hd.

If you were wondering. No. I didn’t just get this all off the top of my head. You can read more about friendly numbers at wolfram, or wikipedia, and here is where the full list of friendly numbers came from. Personally I find amicable pairs more interesting but then, who really cares about that?


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