Glossary of Numbers

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Abundant || Amicable Pairs || Cube || Cute || Deficient || Figurate || Happy || Narcissistic || Palindromic || Perfect || Polygonal || Proper Divisors || Semiperfect || Sociable || Square || Tetrahedral || Triangular || Weird

### 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.

### Cube Numbers

Cube numbers are the result of multiplying a number by itself twice: 13 = 1, 23 = 8, 33 = 27, and so on. (The small 3 means 'cubed' and in e-mail we write it ^3, so that 2^3 is 'two cubed'.)

The cube of 4 is 64, and working backward, we say the cube root of 64 is 4.

If you use cube-shaped blocks to build a larger cube, the number of blocks you need is a cube number. For example, if you want to build a 10-inch cube using 1-inch cube blocks you will need 1000 blocks, the cube of 10.

### 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.

### Figurate Numbers

The number of dots in an arrangement of equally spaced points is a figurate number. Here's an example:

``` 1        3        5       7

*        *        *       *
**       *       *
***     *
****
```

### 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.

### Palindromic Numbers

A palindrome is a word that's the same read either forward or backward, such as noon or kayak. Palindromic numbers, like 88 and 1540451, have the same digits forward and backward.

There's a simple way to turn most numbers into palindromic numbers:

 Pick a number: Reverse its digits: Add them together: Repeat the process until you get a palindromic number. 19 + 91  110 +011  121

Nobody knows whether or not this works for every number. People have used computers to try the flip-and-add process on 196 nearly ten million times, without finding a palindrome-- but it might still be possible. We do know that it won't work for every number written in every base: try 10110 in base 2.

At 8:02 P.M. on Wednesday, February 20th, 2002, time (for sixty seconds only) read in perfect symmetry:

20:02, 20/02, 2002   (200,220,022,002)

It will happen again at 9:12 P.M. on Dec. 21, 2112:   21:12, 21/12, 2112   (211,221,122,112).

### 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.

### Polygonal Numbers

A polygonal number is the number of equally spaced dots needed to draw a polygon. (A polygonal number is a special type of figurate number.) Sequences of polygonal numbers are based on nested polygons. Here's one example:

```1       6           15
**           ***
*       *  *         **  *
**         *  *  *
**  *
***
```

There are many different kinds of polygonal numbers, beginning with square and triangular numbers.

### Proper Divisors

The divisors of an integer are the numbers that it can be divided by without leaving a remainder. For instance, the divisors of 12 are {1, 2, 3, 4, 6, 12}. (Divisors are also called factors.) The proper divisors of a positive integer are all of the divisors less than the integer you started with. Thus, the proper divisors of 12 are {1, 2, 3, 4, 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": 12 = 1, 22 = 4, 32 = 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.

The number of evenly spaced dots needed to make a square is a square number. This is just one kind of polygonal number. Here are some pictures of the first few square numbers:

```*    * *    * * *     * * * *      * * * * *
1    * *    * * *     * * * *      * * * * *
4     * * *     * * * *      * * * * *
9       * * * *      * * * * *
16         * * * * *
25
```

### Tetrahedral Numbers

Tetrahedral numbers are one kind of figurate number. They are found by counting the number of evenly spaced points needed to build a tetrahedron. Tetrahedra are pyramids with triangular bases.

### Triangular Numbers

A triangular number is the number of dots needed to draw a triangle. This is one kind of polygonal number. Here is a picture of the first few triangular numbers:

```
*      *        *          *             *
1     * *      * *        * *           * *
3      * * *      * * *         * * *
6       * * * *       * * * *
10         * * * * *
15
```

The formula for the nth triangular number is T(n) = n (n+1)/2.

### Weird Numbers

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