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 540
- 540 is the smallest number whose divisors contain every digit at least twice
- 540 is the smallest positive heptagonal number that is also a half of a pentagonal number
- 540 is the smallest number n that is a concatenation of two other numbers s and t such that n is the product of the sums of divisors of s and t (540 = 6*90 = σ(5)*σ(40))
- 540 is the smallest number n such that Fibonacci(n) is divisible by n, n+1, n+2, and n+3
Common Properties of 540
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 untouchable numbers are those that are not the sum of the proper divisors of any number.