4 8 15 16 23 42: Numbers ‘Gossip’ [Lost]
February 21, 2010 – 2:20 pm
The Re-up of Wired magazine came yesterday. And with it the revelation of NumberGossip.com offering: “Enter a number and I’ll tell you everything you wanted to know about it but were afraid to ask.”
What better sort of litmus/acid test than to let the site gorge on THE numbers — 4 8 15 16 23 42 — of TV’s “Lost.”
4
Unique Properties of 4
4 is the only compositorial square
4 is the only positive number that is both the sum and the product of the same two integers
4 is the order of the smallest non-cyclic group (submitted by Sam Steingold)
Every positive integer is the sum of at most 4 squares
4 is the smallest number of colors sufficient to color any planar map
4 is the only number in the English language for which the number of letters in its name is equal to the number itself
4 is the only composite number that is equal to the sum of its prime factors
4 is the only composite number n which doesn’t divide (n-1)!
Rare Properties of 4
Cake
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.
2, 4, 8, 15, 26, …
Compositorial
The n-th compositorial is the product of the first n composite numbers.
Compositorial numbers are factorials divided by primorials.
4, 24, 192, 1728, …
Narcissistic
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.
1, 2, 3, 4, 5, 6, 7, …
Power of 2
A number is a power of 2 if it is 2 to some power.
1, 2, 4, 8, 16, 32, …
Square
The number n is a square if it is the square of an integer.
1, 4, 9, 16, 25, …
Tetrahedral (Pyramidal)
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.
1, 4, 10, 20, 35, …
Common Properties of 4
Composite
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
4, 6, 8, 9, …
Deficient
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.
1, 2, 3, 4, 5, 7, 8, …
Even
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.
2, 4, 6, 8, 10, …
Lazy caterer
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.
2, 4, 7, 11, 16, …
Odious
The number n is odious if it has an odd number of 1′s in its binary expansion.
Guess what evil numbers are.
1, 2, 4, 7, 8, 11, …
Palindrome
A palindrome is a number that reads the same forward or backward.
1, 2, 3, 4, 5, 6, 7, …
Powerful
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.
1, 4, 8, 9, 16, …
Practical
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
1, 2, 4, 6, 8, 12, …
Smith (Joke)
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 :-) ?
4, 22, 27, 58, …
Ulam
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.
…, 2, 3, 3, 4, 4, 6, 6, 8, …
8
Unique Properties of 8
8 is the only composite cube in the Fibonacci sequence
8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra
8 is the smallest number (except 1) which is equal to the sum of the digits of its cube
Rare Properties of 8
Cake
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.
2, 4, 8, 15, 26, 42, …
Cube
The number n is a cube if it is the cube of an integer.
1, 8, 27, 64, 125, …
Fibonacci
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.
…, 2, 3, 5, 8, 13, 21, 34, …
Narcissistic
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.
…, 5, 6, 7, 8, 9, 153, 370, …
Power of 2
A number is a power of 2 if it is 2 to some power.
1, 2, 4, 8, 16, 32, 64, …
Common Properties of 8
Composite
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
4, 6, 8, 9, 10, 12, …
Deficient
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.
…, 4, 5, 7, 8, 9, 10, 11, …
Even
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.
2, 4, 6, 8, 10, 12, 14, …
Odious
The number n is odious if it has an odd number of 1′s in its binary expansion.
Guess what evil numbers are.
…, 2, 4, 7, 8, 11, 13, 14, …
Palindrome
A palindrome is a number that reads the same forward or backward.
…, 5, 6, 7, 8, 9, 11, 22, …
Powerful
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.
1, 4, 8, 9, 16, 25, …
Practical
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
…, 2, 4, 6, 8, 12, 16, 18, …
Ulam
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.
…, 4, 6, 6, 8, 8, 11, 11, 13, …
15
Unique Properties of 15
15 is the smallest emirpimes
15 is the smallest composite cyclic number, that is number n with the property that there is only one group of order n
15 is the magic constant of the unique order-3 normal magic square
15 is the number of letters in the words “uncopyrightable” and “dermatoglyphics”, which are the only two longest words there are without repeating a letter
Rare Properties of 15
Cake
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.
2, 4, 8, 15, 26, 42, 64, …
Common Properties of 15
Composite
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
…, 10, 12, 14, 15, 16, 18, 20, …
Deficient
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.
…, 11, 13, 14, 15, 16, 17, 19, …
Evil
The number n is evil if it has an even number of 1′s in its binary expansion.
Guess what odious numbers are.
…, 9, 10, 12, 15, 17, 18, 20, …
Lucky
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.
…, 7, 9, 13, 15, 21, 25, 31, …
Odd
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.
…, 9, 11, 13, 15, 17, 19, 21, …
Square-free
A number is said to be square-free if its prime decomposition contains no repeated factors.
…, 11, 13, 14, 15, 17, 19, 21, …
Triangular
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.
…, 3, 6, 10, 15, 21, 28, 36, …
16
Unique Properties of 16
16 is the number of vertices of a tesseract
16 is the only number of the form xy=yx with different x and y
16 is the smallest prime power of a prime power of a prime
16 is the base of the hexadecimal number system, which is used extensively in computer science
Rare Properties of 16
Power of 2
A number is a power of 2 if it is 2 to some power.
…, 2, 4, 8, 16, 32, 64, 128, …
Square
The number n is a square if it is the square of an integer.
1, 4, 9, 16, 25, 36, 49, …
Common Properties of 16
Composite
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
…, 12, 14, 15, 16, 18, 20, 21, …
Deficient
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.
…, 13, 14, 15, 16, 17, 19, 21, …
Even
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.
…, 10, 12, 14, 16, 18, 20, 22, …
Lazy caterer
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.
…, 4, 7, 11, 16, 22, 29, 37, …
Odious
The number n is odious if it has an odd number of 1′s in its binary expansion.
Guess what evil numbers are.
…, 11, 13, 14, 16, 19, 21, 22, …
Powerful
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.
…, 4, 8, 9, 16, 25, 27, 32, …
Practical
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
…, 6, 8, 12, 16, 18, 20, 24, …
Ulam
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.
…, 11, 13, 13, 16, 16, 18, 18, 26, …
23
Unique Properties of 23
23 is the smallest group of people where there is more than a 50% chance that 2 people will share the same birthday (day and month, not year)
23 is the smallest isolated prime, i.e., not an element of a set of twin primes
23 is the smallest prime whose reversal is a power: 32 = 25
23 is the only prime p such that p! is p digits long
23! is the least pandigital factorial, that is it contains all the digits 0 through 9 at least once
A web page about 23: 23 Enigma in wikipedia
23 is the smallest prime p such that the ring of integers in the cyclotomic field of pth roots of unity does not have unique factorization (submitted by Qiaochu Yuan)
Common Properties of 23
Deficient
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.
…, 19, 21, 22, 23, 25, 26, 27, …
Evil
The number n is evil if it has an even number of 1′s in its binary expansion.
Guess what odious numbers are.
…, 17, 18, 20, 23, 24, 27, 29, …
Happy
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.
…, 10, 13, 19, 23, 28, 31, 32, …
Odd
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.
…, 17, 19, 21, 23, 25, 27, 29, …
Prime
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.
…, 13, 17, 19, 23, 29, 31, 37, …
Square-free
A number is said to be square-free if its prime decomposition contains no repeated factors.
…, 19, 21, 22, 23, 26, 29, 30, …
42
Unique Properties of 42
The number 42 is The Ultimate Answer to The Ultimate Question of Life, the Universe and Everything
42 is the number of spots on a pair of dice (submitted by Ken Knowlton)
42 is the alphanumeric value of FIVE
Number 42 on the web:
The Answer to Life, the Universe, and Everything
Comparative frequency analysis of 42 and 47 against other numbers from Thoreaulylazy’s blog
42 is the smallest abundant odious number
Rare Properties of 42
Cake
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.
…, 8, 15, 26, 42, 64, 93, 130, …
Catalan
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.
…, 2, 5, 14, 42, 132, 429, 1430, …
Pronic (Heteromecic)
The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
…, 12, 20, 30, 42, 56, 72, 90, …
Common Properties of 42
Abundant
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.
…, 30, 36, 40, 42, 48, 54, 56, …
Composite
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
…, 38, 39, 40, 42, 44, 45, 46, …
Even
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.
…, 36, 38, 40, 42, 44, 46, 48, …
Odious
The number n is odious if it has an odd number of 1′s in its binary expansion.
Guess what evil numbers are.
…, 37, 38, 41, 42, 44, 47, 49, …
Practical
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
…, 32, 36, 40, 42, 48, 54, 56, …
Square-free
A number is said to be square-free if its prime decomposition contains no repeated factors.
…, 38, 39, 41, 42, 43, 46, 47, …
~
Let’s go back to just an overview of their Rare and Common Properties for any common threads:
Common Properties of 4
Rare Properties of 15
