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.
From the Web:
Cubes, Richard Phillips, Univ. of Nottingham
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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.
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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
* * * *
** * *
*** *
****
The points can be arranged in one, two, three, or even more dimensions.
There are many different kinds of figurate numbers, such as
polygonal and tetrahedral numbers.
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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.
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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.
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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).
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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}.
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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.
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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.
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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
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Glossary of Numbers 
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