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 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
- 16 is the smallest short leg in a primitive Pythagorean triangle whose three side lengths are composite integers
- 16 is the smallest number which is the sum of two distinct odd primes in two ways: 16 = 3 + 13 = 5 + 11
- 16 is the only 2-digit number n such that nn ends with two copies of n: 1616 = 18446744073709551616
- 16 is the largest number n such that every set of n consecutive integers contain a number which is relatively prime to all the others
Rare Properties of 16
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.
Common Properties of 16
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.
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.
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.