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 64
- 64 is the number of squares on a chessboard
- 64 is the smallest power of 2 with no prime neighbor
- 64 is the smallest non-trivial square and cube at the same time
- 64 is alphanumeric value of ZERO
- 64 is the difference of the first pair of amicable numbers: 284 − 220
Rare Properties of 64
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 number n is a cube if it is the cube of an integer.
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 64
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 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.