Practical numbers

What are practical numbers? Are they defined by housewife-like mathematicians as numbers are useful for every purpose? Of course not, else they would really be housewives. Practical numbers are really numbers such that numbers smaller than it can be represented as sums of different divisors of that number.

What do you mean, I hear, right from my desktop. To illustrate, let me use my index number, 28. Its divisors are 1, 2, 4, 7, 14 and 28. Hence, to show that 28 is indeed a practical number, we have to show that by adding up some of the numbers from the set {1, 2, 4, 7, 14, 28} , we can get any number we like from 1 to 27. Let's say I want numbers from 10 to 20. Let's see how it can be done:

10 = 1 + 2 + 7
11 = 4 + 7
12 = 1 + 4 + 7
13 = 2 + 4 + 7
14 = 1 + 2 + 4 + 7 (alternatively you can just put 14 = 14 )
15 = 1 + 14
16 = 2 + 14
17 = 1+ 2 + 14
18 = 4 + 14
19 = 1 + 4 + 14
20 = 2 + 4 + 14

If we continue the list both ways, we find out that we can represent numbers from 1 to 27 as sums of the divisors of 28. Hence, we can conclude that 28 is indeed a practical number. Well, 28 might not practical to you, but it is to me!