Abundant and Deficient Numbers and the Sigma Function

MATH 035 Penn State University Dr. James Sellers

Handout: Abundant and Deficient Numbers and the Sigma Function

We’ve spent several lessons on the perfect numbers – their historical significance, their “structure” (at least in the case of the even perfect numbers), and various results related to them. In this lesson, we study close relatives of the perfect numbers known as abundant numbers and deficient numbers.

Definition: A positive integer n is called an abundant number if the sum of all the proper divisors of n is greater than n.

Definition: A positive integer n is called a deficient number if the sum of all the proper divisors of n is less than n.

Question 1: How are these definitions related to the definition of a perfect number?

It is not exactly clear when these numbers first appeared in the literature. But we know that they date back at least to Nicomachus (60-120 A.D.) and his book Introduction to Arithmetic . In that text, Nicomachus speaks of abundant and deficient numbers in very “moral” or “judgmental” language. Note the following material from Nicomachus’ biography at the MacTutor History of Mathematics website:

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

He then continues his description of abundant numbers as resembling an animal:-

... with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands....

while a deficient number is like an animal:- ... with a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

With this in mind, it would be great to answer a few questions about abundant and deficient numbers including the following:

Question 2: Are there any “well-known” families of numbers which are either always abundant or always deficient?

Question 3: In a slightly weaker sense, how many abundant numbers are there? How many deficient numbers are there? Are they all only even (as is currently the case with the known perfect numbers)?

With these questions in mind, please characterize the numbers below as abundant, perfect, or deficient by placing the letter A, P, or D next to each one. Use the definitions above to determine these characterizations.

11 ______21 ______31 ______

2 ______12 ______22 ______32 ______

3 ______13 ______23 ______33 ______

4 ______14 ______24 ______34 ______

5 ______15 ______25 ______35 ______

6 ______16 ______26 ______36 ______

7 ______17 ______27 ______37 ______

8 ______18 ______28 ______38 ______

9 ______19 ______29 ______39 ______

10 ______20 ______30 ______40 ______

Now that you have these numbers characterized, can you make any conjectures about answers to Questions 2 and 3 above? The work above motivates us to now consider the function often called the “sigma function”, which is typically denoted by σ (n ) , and simply records the sum of ALL the divisors of n (including n itself).

So let’s make sure we see how σ (n ) relates to abundant, perfect, and deficient numbers:

• If σ (n )> 2 n , then we say that n is a(n) ______number.

• If σ (n )= 2 n , then we say that n is a(n) ______number.

• If σ (n )< 2 n , then we say that n is a(n) ______number.

It would be extremely nice to have some known facts which would help us calculate the value of σ (n ) very quickly. Thankfully, such facts exist. Here are the main ones we need to remember:

1. σ (p )= p + 1 whenever p is prime

p j+1 −1 2. σ (p j ) = whenever p is prime p −1

3. σ(M× ) = σ ()() M × σ if M and N are relatively prime (this is called multiplicativity )

Fact 3 above is especially helpful for finding σ (n ) if the prime factorization of n is know. These facts are basically all we need to calculate the sigma function for ANY positive integer. Let’s do some examples on the board now to illustrate these facts.