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 720
- 720 is the largest factorial to contain all different digits
- 720 is the smallest number with 30 divisors
- 720 is the smallest number n such that nτ(n) > prime(3n)
- 720 is the smallest number n such that τ(n)φ(n) > prime(n)
- 720 is the smallest number n such that σ(n)2 is greater than 3σ(n2)
- 720 is the smallest number n such that prime(n) ± n and prime(n) ± 2n are all primes
- 720 is the smallest number n where either n or n+1 is divisible by the numbers from 1 to 10
- 720 is the product of the sides and the perimeter of the smallest Pythagorean triangle
- 720 is the smallest positive number differing from the next 3 greater squares by squares (720 = 272 - 32 = 282 - 82 = 292 - 112)
Rare Properties of 720
The n-th factorial is the product of the first n natural numbers.
The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.
Common Properties of 720
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.
The number n is called an apocalyptic power if 2n contains the consecutive digits 666 (in decimal).
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 next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.