Number Gossip Properties

Abundant

Definition: The number n is abundant if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

First ten: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54

There are 2487 abundant numbers below 10,000.

Amicable

Definition: The number n is amicable if it belongs to an amicable pair. Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.

It all started with perfect numbers which are amicable with themselves. Those numbers adopted social virtues and qualities; for the parts of each of them have the power to generate the other. See also sociable numbers.

First ten: 220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368

There are 10 amicable numbers below 10,000.

Apocalyptic power

Definition: The number n is called an apocalyptic power if 2n contains the consecutive digits 666 (in decimal).

First ten: 157, 192, 218, 220, 222, 224, 226, 243, 245, 247

There are 6485 apocalyptic powers below 10,000.

Aspiring

Definition: The number n is called an aspiring number if its aliquot sequence terminates in a perfect number, and it is not a perfect number itself.

First ten (known): 25, 95, 119, 143, 417, 445, 565, 608, 650, 652

There are 89 known aspiring numbers below 10,000.

Automorphic (Curious)

Definition: The number n is called an automorphic number if (the decimal expansion of) n2 ends with n. These numbers are also called curious.

It is curious, how for a k-digit automorphic number n there is another automorphic number -- 10k + 1 - n. For this to work with n=1, you have to treat 1 as a zero-digit number.

First ten: 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376

There are 8 automorphic numbers below 10,000.

Cake

Definition: The n-th cake number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.

Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.

First ten: 2, 4, 8, 15, 26, 42, 64, 93, 130, 176

There are 39 cake numbers below 10,000.

Carmichael

Definition: The composite integer n is a Carmichael number if bn-1 = 1 (mod n) for every integer b which is relatively prime with n.

Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.

First ten: 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341

There are 7 Carmichael numbers below 10,000.

Catalan

Definition: The n-th Catalan number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).

There are many ways Catalan numbers can be interpreted; there are some cool pictures here and the Wikipedia article is very good.

First ten: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796

There are 9 Catalan numbers below 10,000.

Composite

Definition: A positive integer greater than 1 that is not prime is called composite.

Composite numbers are opposite to prime numbers.

First ten: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18

There are 8769 composite numbers below 10,000.

Compositorial

Definition: The n-th compositorial is the product of the first n composite numbers.

Compositorial numbers are factorials divided by primorials.

First ten: 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800

There are 4 compositorials below 10,000.

Cube

Definition: The number n is a cube if it is the cube of an integer.

First ten: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

There are 21 cube numbers below 10,000.

Deficient

Definition: The number n is deficient if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

First ten: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11

There are 7508 deficient numbers below 10,000.

Even

Definition: A number is even if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

First ten: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

There are 4999 even numbers below 10,000.

Evil

Definition: The number n is evil if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

First ten: 3, 5, 6, 9, 10, 12, 15, 17, 18, 20

There are 4999 evil numbers below 10,000.

Factorial

Definition: The n-th factorial is the product of the first n natural numbers.

The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.

First ten: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800

There are 7 factorials below 10,000.

Fibonacci

Definition: Fibonacci numbers are numbers that form the Fibonacci sequence. The Fibonacci sequence is defined as starting with 1, 1 and then each next term is the sum of the two preceding ones.

Fibonacci numbers are very common in nature. For example, a pineapple has 8 spirals if you count one way, and 13 if you count the other way.

First ten: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

There are 19 different Fibonacci numbers below 10,000.

Google

Definition: The n-th Google number is the first n-digit prime found in the decimal expansion of e.

They are named Google numbers because of the unusual hiring ad that Google put up.

First ten: 2, 71, 271, 4523, 74713, 904523, 2718281, 72407663, 360287471, 7427466391

There are 4 Google numbers below 10,000.

Happy

Definition: One can take the sum of the squares of the digits of a number. Those numbers are happy for which iterating this operation eventually leads to 1.

First ten: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44

There are 1441 happy numbers below 10,000.

Hungry

Definition: The k-th hungry number is the smallest number n such that 2^n contains the first k digits of the decimal expansion of pi.

They are named hungry numbers because they try to eat as much "pi" as possible.

First ten: 5, 17, 74, 144, 144, 2003, 2003, 37929, 82810, 161449

There are 7 hungry numbers below 10,000.

Lazy caterer

Definition: The n-th lazy caterer number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

First ten: 2, 4, 7, 11, 16, 22, 29, 37, 46, 56

There are 140 lazy caterer numbers below 10,000.

Lucky

Definition: To build the lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, ... . The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, ... . The next number remaining is 9, so delete every ninth number, etc.

Those numbers were lucky they weren't crossed out.

First ten: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33

There are 1118 lucky numbers below 10,000.

Mersenne

Definition: A number of the form 2p - 1 is called a Mersenne number if p is prime.

It was believed many years ago, that all Mersenne numbers are prime. This is not so, thus there is a separate entry for Mersenne prime numbers.

First ten: 3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911

There are 6 Mersenne numbers below 10,000.

Mersenne prime

Definition: A Mersenne number which is also prime is called a Mersenne prime.

The drive to find prime numbers among Mersenne numbers supplies humanity with the largest known prime numbers.

First ten: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111

There are 5 Mersenne primes below 10,000.

Narcissistic

Definition: A k-digit number n is called narcissistic if it is equal to the sum of k-th powers of its digits. They are also called Plus Perfect numbers.

First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153

There are 16 narcissistic numbers below 10,000.

Odd

Definition: A number is odd if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

First ten: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

There are 5000 odd numbers below 10,000.

Odious

Definition: The number n is odious if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

First ten: 1, 2, 4, 7, 8, 11, 13, 14, 16, 19

There are 5000 odious numbers below 10,000.

Palindrome

Definition: A palindrome is a number that reads the same forward or backward.

First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11

There are 198 palindromic numbers below 10,000.

Palindromic prime

Definition: A palindromic prime is a prime which is a palindrome.

In base 2 Mersenne primes are palindromic primes.

First ten: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191

There are 20 palindromic primes below 10,000.

Pentagonal

Definition: Pentagonal numbers are of the form n(3n - 1)/2.

Pentagonal numbers are to pentagons what triangular numbers are to triangles and square numbers are to squares.

First ten: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145

There are 81 pentagonal numbers below 10,000.

Perfect

Definition: The number n is perfect if the sum of all its positive divisors except itself is equal to n.

Less than perfect numbers are called deficient, too perfect -- abundant.

First ten: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216

There are 4 perfect numbers below 10,000.

Power of 2

Definition: A number is a power of 2 if it is 2 to some power.

First ten: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512

There are 14 powers of 2 below 10,000.

Powerful

Definition: An integer n is powerful if for every prime p dividing n, p2 also divides n.

How much power? They all can be written as a2 b3.

First ten: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49

There are 184 powerful numbers below 10,000.

Practical

Definition: The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.

First ten: 1, 2, 4, 6, 8, 12, 16, 18, 20, 24

There are 1455 practical numbers below 10,000.

Prime

Definition: A prime is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

First ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

There are 1229 primes below 10,000.

Primorial

Definition: The p-primorial is the product of all primes less than or equal to p. It is sometimes denoted by p#.

Compare to compositorials and factorials.

First ten: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230

There are 5 primorials below 10,000.

Pronic (Heteromecic)

Definition: The number is called pronic if it is the product of two consecutive numbers.

They are twice triangular numbers.

First ten: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110

There are 99 pronic numbers below 10,000.

Repunit

Definition: A repunit is an integer in which every digit is one.

The term repunit comes from combining "repeated" and "unit".

First ten: 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111

There are 4 repunits below 10,000.

Smith (Joke)

Definition: A composite number is called a Smith number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?

First ten: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265

There are 376 Smith numbers below 10,000.

Sociable

Definition: An aliquot sequence is formed by taking an integer, adding all of its divisors other than itself, and then repeating this process with the sum. The numbers for which this process returns to the starting point after more than two steps are called sociable numbers.

The 2-cycles are the amicable pairs and the 1-cycles are the perfect numbers. For some numbers it is very difficult to compute the aliquot sequence. The smallest number whose sequence has not been completely computed is 276.

First ten (known): 12496, 14264, 14288, 14316, 14536, 15472, 17716, 19116, 19916, 22744

There are no known sociable numbers below 10,000.

Square

Definition: The number n is a square if it is the square of an integer.

First ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

There are 99 squares below 10,000.

Square-free

Definition: A number is said to be square-free if its prime decomposition contains no repeated factors.

First ten: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14

There are 6083 square-free numbers below 10,000.

Tetrahedral (Pyramidal)

Definition: A tetrahedral number is the number of balls you can put in a triangular pyramid.

This is the space generalization of triangular and square numbers.

First ten: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220

There are 38 tetrahedral numbers below 10,000.

Triangular

Definition: If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a triangle number.

Compare to square, pentagonal and tetrahedral numbers.

First ten: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

There are 140 triangular numbers below 10,000.

Twin

Definition: A prime number is called a twin prime if there exists another prime number differing from it by 2.

First ten: 3, 5, 7, 11, 13, 17, 19, 29, 31, 41

There are 409 twin primes below 10,000.

Ulam

Definition: The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.

First ten: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18

There are 827 Ulam numbers below 10,000.

Undulating

Definition: Undulating numbers are numbers of the form abababab... in base 10.

This property is significant starting from 3-digit numbers, so we will not consider numbers below 100.

First ten: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191

There are 180 undulating numbers below 10,000.

Untouchable

Definition: The untouchable numbers are those that are not the sum of the proper divisors of any number.

First ten: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188

There are 1212 untouchable numbers below 10,000.

Vampire

Definition: The number n is called a vampire number if there exists a factorization of n using n's digits.

First ten: 126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530

There are 15 vampire numbers below 10,000.

Weird

Definition: The number n is called a weird number if it is abundant but is not the sum of any subset of its proper factors.

First ten: 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792

There are 7 weird numbers below 10,000.