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Abundant || Amicable Pairs || Cube || Cute || Deficient || Figurate || Happy || Narcissistic ## Abundant NumbersA 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.
Abundant and Deficient Numbers
Abundant Number, from Eric Weisstein's World of Mathematics ## Amicable PairsYou 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. - Amicable Numbers
- Amicable Partners
- Defining Kinds of Numbers
**From the Web:** - Amicable Pair, from Eric Weisstein's World of Mathematics
- Perfect, amicable and sociable numbers, David Moews
- A high-level introduction, with lists of amicable numbers.
## Cube NumbersCube numbers are the result of multiplying a number by itself twice: 1 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.
Cubes, Richard Phillips, Univ. of Nottingham ## Cute NumbersIf 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 NumbersIf 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
Abundant and Deficient Numbers
Deficient Number, from Eric Weisstein's World of Mathematics ## Figurate NumbersThe 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.
Pascal's Triangle: Number Patterns
Figurate Number, from Eric Weisstein's World of Mathematics ## Happy NumbersA 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.
Happy Numbers, Intermath ## Narcissistic NumbersA 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. - Narcissistic Numbers, Weird Numbers, and Fortunate Primes
**From the Web:** - Narcissistic Number, Eric Weisstein's World of Mathematics
- A rigorous introduction, with lists of narcissistic numbers.
- Wild Narcissistic Numbers, Mike Keith
- Narcissistic numbers from unusual functions.
## Palindromic NumbersA 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 There's a simple way to turn most numbers into palindromic numbers:
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). - One way to count the Palindromic Numbers less than 10, 100, and 100 000.
- Another way to count the Palindromic Numbers less than 100 000.
- Making Numbers Into Palindromic Numbers
**From the Web:** - Palindromic Number, Eric Weisstein's World of Mathematics
- A more rigorous definition: see also the Palindromic Number Conjecture and Palindromic Primes.
- Palindromic Numbers and Other Recreational Topics, Patrick De Geest
- Palindromic numbers, triangles, squares, tetrahedra, primes, pronic numbers, Pythagorean triples, and more.
- Digit Reversal Sums Leading to Palindromes, Kevin Brown
- Three Years of Computing, John Walker
- A "quest" to discover whether 196 can be made into a palindrome by the flip-and-add process, with a C program to download.
## Perfect NumbersThe 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. - Perfect Numbers FAQ
- Perfect Numbers - Basics, History
- Perfect Numbers
- How many perfect numbers have been found?
- Perfect Numbers
- What is the sum of the digits of a perfect number? Are there any odd perfects?
- Euclidean Algorithms
- What is the significance of a perfect number?
- Perfect Number, Eric Weisstein's World of Mathematics
- "A Perfect Collaboration," Ivars Petersen
- The early history of perfect numbers.
- Mersenne Primes: History, Theorems, and Lists
- GIMPS: The Great Internet Mersenne Prime Search
## Polygonal NumbersA 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. - Figurate and Polygonal Numbers
- Figurate Numbers
**From the Web:** - Polygonal Number, from Eric Weisstein's World of Mathematics
## Proper DivisorsThe 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}. - Proper Divisor, from Eric Weisstein's World of Mathematics
- Aliquot game
## Semiperfect NumbersA semiperfect number is the sum of
Narcissistic Numbers, Weird Numbers, and Fortunate Primes
Semiperfect Number, Eric Weisstein's World of Mathematics ## Sociable NumbersSociable 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. - Sociable Numbers, Eric Weisstein's World of Mathematics
- Perfect, amicable and sociable numbers, David Moews
- A high-level introduction, with lists of sociable numbers.
## Square NumbersSquare numbers are the result of multiplying a number by itself once. These are the same as the "perfect squares": 1 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 - Pattern of Squares
- Sum of First n Odd Numbers
- Sum of n Odd Numbers (This one uses induction.)
- Odd Digits of Square Numbers
**From the Web:** - Square and Triangular Numbers, Alexander Bogomolny
- The same site proves that there are an infinite amount of numbers that are both square and triangular.
- Squares, Richard Phillips, Univ. of Nottingham
## Tetrahedral NumbersTetrahedral 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. - Summing Triangle Numbers (Dr. Math archives)
- Pascal's Triangle: Tetrahedral Numbers
**From the Web:** - Tetrahedral Number, Eric Weisstein's World of Mathematics
## Triangular NumbersA 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. - Triangular Numbers
- Why is 1 a triangular number?
- Formula for Triangular Numbers
- Triangular Numbers
- Identifying triangular numbers, and one way to use them.
- Summing Triangle Numbers
- Triangular Numbers
- How do I prove that there is an infinite number of triangular numbers that are equal to a square number?
- Pascal's Triangle: Triangular Numbers
- Triangular Number Identities I
- An Algebra Problem of the Week.
- Mrs. Pascalini's Garden
- A Middle School Problem of the Week.
- Triangular Number, from Eric Weisstein's World of Mathematics
- Square and Triangular Numbers, Alexander Bogomolny
- The same site proves that the number of numbers that are both square and triangular is infinite.
## Weird NumbersA number is weird if it is abundant without being semiperfect; 70 is the first weird number.
Narcissistic Numbers, Weird Numbers, and Fortunate Primes
Weird Number, from Eric Weisstein's World of Mathematics - Ursula Whitcher, for the Math Forum |

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