Number Gossip
(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)
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 for which the number of letters in its English 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)!
- The tetrahedron has 4 Vertices and 4 daces
- 4 is the largest number such that its divisors plus 1 are all prime
- 4 is the smallest composite number
- 4 is the smallest non-Fibonacci number
Rare Properties of 4
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.
The n-th compositorial is the product of the first n composite numbers.
Compositorial numbers are factorials divided by primorials.
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.
A number is a power of 2 if it is 2 to some power.
The number n is a square if it is the square of an integer.
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.
Common Properties of 4
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
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.
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.
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.
The number n is odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
A palindrome is a number that reads the same forward or backward.
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.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
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 :-) ?
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.