Why are these number pairs not Amicable Numbers?

Question

It is a known fact that (220, 284) is the smallest pair of amicable numbers. That proves that I don't understand at least one part of finding amicable pairs...

Please explain to me where I fall down:

Suppose I have the numbers 2 and 3. The proper divisors of 2 is 1. The proper divisors of 3 is 1 too. Am I correct here? So, why isn't (2,3) a legitimate amicable pair?

And what about (48 , 92) ? The proper divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 The proper divisors of 92 are 1, 2, 4, 23, 46

Both sums equal to 76.

So where do I fail to understand amicable numbers?

Thanks

Answer 1

It looks like you don't understand the definition of amicable numbers.

$220$ and $284$ are amicable numbers because the sum of the proper divisors of $220$ is $$1+2+4+5+10+11+20+22+44+55+110=284$$ while the sum of the proper divisors of $284$ is $$1+2+4+71+142=220.$$

If $\sigma(a)-a=b$ and $\sigma(b)-b=a$ and $a\ne b,$ them $(a,b)$ is called an amicable pair.

If $\sigma(a)-a=b$ and $\sigma(b)-b=a$ and $a=b,$ then $a$ is called a perfect number.

Answer 2

Amicable pairs aren't pairs where each number's sum of proper divisors is the same. An amicable pair is where each number's proper divisors sum to the other number. So the sum of proper divisors for 220 is 284, and vice versa.