I've been getting mail all day asking me to explain something
that appeared in today's XKCD comic. Yes, I've been reduced to explaining geek comics to my readers. I suppose that there are worse fates. I just can't
think of any. 🙂
But seriously, I'm a huge XKCD fan, and I don't mind explaining interesting things no matter what the source. If you haven't read today's
comic, follow the link, and go look. It's funny, and you'll know what
people have been asking me about.
The comic refers to friendly numbers. The question,
obviously, is what are friendly numbers?
First, we define something called a divisors function over the integers, written σ(n). For any integer, there's a set of integers that divide
into it. For example, for 4, that's 1, 2, and 4. For 5, it's just 1 and 5. And for 6, it's 1, 2, 3, 6. The divisors function, σ(n) is the sum of all of the divisors of n. So
$ sigma(4)=8, sigma(5)=6, sigma(6)=12.$
For each integer, there is a characteristic ratio, defined
using the divisors function. For the integer n, the characteristic
is the ratio of the divisors function over the the number itself: σ(n)/n. So the characteristic ratio of 4 is 7/4; for 6, it's
For any characteristic ratio, the set of numbers that share that characteristic are friendly with each other. A friendly number is,
therefore, any integer that shares its characteristic ratio with at least one other integer. If an integer isn't friendly, then it's called a solitary number. 1, 2, 3, 4, and 5 are all solitary numbers. 6 is
friendly with 28 (1+2+4+7+14+28/28 = 56/28 = 2).
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