jmc

Prime factorize $284=2^2\cdot 71$. The sum of the proper divisors of $284$ is $$\begin{array}{rl}1+2+2^2+71+2\cdot 71& =(1+2+2^2)(1+71)-284\\ & =220\\ & =2^2\cdot 5\cdot 11.\end{array}$$Here we have used the observation that multiplying out $(1+2+2^2)(1+71)$ by distributing yields an expression which is the sum of all $6$ factors of $284.$ Applying this observation again, we find that the sum of the proper divisors of $220$ is $$(1+2+2^2)(1+5)(1+11)-220=7\cdot 6\cdot 12-220=\boxed{284}.$$