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 1680
- 1680 is the smallest number with 40 divisors
- 1680 is the smallest area of a triangle with sides, area, and two medians being integers
- 1680 is the smallest number n such that 2n − 1 is less than σ(n) &minus σ(n-2). Equivalently, 1680 is the smallest n such that the sum of non-divisors of n is less than the sum of non-divisors of n-2
- 1680 is the smallest number n such that 2n - φ(x) = σ(x)/2
Common Properties of 1680
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.
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.