"Happy","Cute", "Sociable", "Weird" And Other Special Numbers

In a previous post (See http://raffysaldana.blogspot.com/2007/08/palindromes-and-palindromic-numbers.html) I talked about a special number called a palindromic number. In this article, I will discuss other special numbers...


“Abundant”, “amicable”, “cute”, “deficient”, “happy”, “narcissistic”, “perfect”, “semiperfect”, “sociable”, “square”, “weird”.

These adjectives normally pertain to people or, somehow, they are related to people.

But did you know that these adjectives also pertain to numbers?

In mathematics, there is such a field called number theory. Number theorists are mathematicians that study, among others, the characteristics of special numbers.

Here are some excerpts from a website “Math Forum@Drexel” ( http://mathforum.org/dr.math/faq/faq.number.glossary.html) on some kind of special numbers:


“Abundant Numbers

A number is abundant if the sum of its proper divisors is greater than the number itself. For example, the proper divisors of 24 are {1, 2, 3, 4, 6, 8, 12} and 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, so 24 is abundant."


"Amicable Pairs

You might think of an amicable pair as two numbers that are best friends. The sum of the proper divisors of the first number is the second number, and if you add up the proper divisors of the second number, you get the first number.

Here's an example. One amicable pair is 2620 and 2924. The proper divisors of 2620 are {1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310}. Their sum is 1 + 2 + 4 + 5 + 10 + 20 + 131 + 262 + 524 + 655 + 1310 = 2924. Next we check whether 2924's proper divisors add up to 2620. 2924's proper divisors are {1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462}. 1 + 2 + 4 + 17 + 34 + 43 + 68 + 86 + 172 + 731 + 1462 = 2620, so the pair of numbers really is amicable."

"Cute Numbers

If a square can be cut into n squares of at the most two different sizes, then n is called a cute number. For example, 4 and 10 are cute numbers."

"Deficient Numbers

If the sum of a number's proper divisors is less than the original number, it is called a deficient number. For instance, 16 is deficient. The proper divisors of 16 are {1, 2, 4, 8}, but 1 + 2 + 4 + 8 = 15."


"Happy Numbers

A happy number is a number for which the sum of the squares of the digits eventually equals 1.

For instance, 203 is happy:

2^2 + 0^2 + 3^2 = 13
1^2 + 3^2 = 10
1^2 + 0^2 = 1.

Numbers that are not happy, such as 16, are called unhappy numbers."


"Narcissistic Numbers

A narcissistic person is only interested in himself; a narcissistic number might seem a little self-centered, too. A narcissistic number is an integer equal to an expression that uses the same digits. For example, 36 = 3! * 6. Sometimes a narcissistic number is defined as a number equal to the sum of its digits raised to a certain power, or, more specifically, as an n-digit number equal to the sum of its digits raised to the nth power. For instance, 371 is narcissistic because 3^3 + 7^3 + 1^3 = 371, and 9474 is narcissistic because 9^4 + 4^4 + 7^4 + 4^4 = 9474."


"Perfect Numbers

The numbers that divide evenly into an integer are called its divisors. For example, the divisors of 6 are {1, 2, 3, 6}. Proper divisors are the divisors less than the integer you started with: the proper divisors of 6 are {1, 2, 3}. A number is perfect if it is equal to the sum of its proper divisors. 6 is perfect, because 1 + 2 + 3 = 6."


"Semiperfect Numbers

A semiperfect number is the sum of some of its proper divisors. For instance, 18 is semiperfect because its proper divisors are {1, 2, 3, 6, 9} and 3 + 6 + 9 = 18. If a semiperfect number is the sum of all of its proper divisors, it is called a perfect number."

"Sociable Numbers

Sociable numbers are like amicable numbers, but they come in larger groups. The proper divisors of the first number in the group add up to the second number, the proper divisors of the second number add up to the third number, and so on. The sum of the proper divisors of the last number in the group is equal to the first number. Sociable numbers tend to be quite large, so they are hard to find without using a computer. One example of a sociable group is 12496, 14288, 15472, 14536, and 14264."

"Square Numbers

Square numbers are the result of multiplying a number by itself once. These are the same as the "perfect squares": 1^2 = 1, 2^2 = 4, 3^2 = 9, and so on. (The small ^2 means 'squared' and in e-mail we write it ^2, so that 2^2 is 'two squared'.)

The square of 5 is 25, and working backward, we say the square root of 25 is 5. "

"Weird Numbers

A number is weird if it is abundant without being semiperfect; 70 is the first weird number."


Cheers,


Raffy
8/17/07 in Quezon City