The X+Y Files

Issue 4

Numbers with Names

Perfect, Abundant or Deficient?

When exploring the properites of a number, it is often a good idea to look at its factors. For each number, N, in the table below, we list all the factors (other than N itself) and the sum, S, of these factors. We can then decide whether the number is perfect, abundant or deficient by comparing S and N. If S = N, then the number is perfect. If S > N , then the number has lots of factors and is called abundant. However, if S < N , it is rather short of factors and is said to be deficient. We find that 6 is perfect, 12, 18 and 20 are abundant, but the other numbers between 2 and 20 are deficient.

Number

Factors

Sum

Type

2 

1

  1

Deficient

3 

1

  1

Deficient

4 

1, 2

  3

Deficient

5 

1

  1

Deficient

6 

1, 2, 3

  6

Perfect

7 

1

  1

Deficient

8 

1, 2, 4

  7

Deficient

9 

1, 3

14

Deficient

10

 1, 2, 5

  8

Deficient

11

 1

  1

Deficient

12

 1, 2, 3, 4, 6

16

Abundant

13

 1

  1

Deficient

14

 1,  2, 7

10

Deficient

15

 1, 3, 5

  9

Deficient

16

 1, 2, 4, 8

15

Deficient

17

 1

  1

Deficient

18

 1, 2, 3, 6, 9

21

Abundant

19

 1

  1

Deficient

20

 1, 2, 4, 5, 10

22

Abundant

Perfect numbers are rare. There is only one 2 digit perfect number. Can you find it? Abundant numbers are easier to find. How many abundant numbers are there between 2 and 100?

We can go further and talk about numbers which are superabundant . An abundant number, N, is called superabundant if the value of S / N is greater than the value obtained for any smaller number. The first superabundant number is 12. What is the next one?