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 1728
- 1728 is the only compositorial cube (submitted by Sergei Bernstein)
Rare Properties of 1728
The n-th compositorial is the product of the first n composite numbers.
Compositorial numbers are factorials divided by primorials.
- 4,
- 24,
- 192,
- 1728,
- 17280,
- 207360,
- 2903040,
- ...
The number n is a cube if it is the cube of an integer.
Common Properties of 1728
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.
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime 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 evil if it has an even number of 1's in its binary expansion.
Guess what odious 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 untouchable numbers are those that are not the sum of the proper divisors of any number.