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 1681
- 1681 (= 412), is smallest composite number generated by the Euler polynomial n2 + n + 41
- 1681 is the only four-digit square not ending in two zeroes such that each of its two 2-digit parts is a square
- 1681 is the smallest number n such that n*prime(n)+(n+1)*prime(n+1) is a perfect square
Rare Properties of 1681
The number n is a square if it is the square of an integer.
Common Properties of 1681
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 odd if it is not divisible by 2.
Numbers that are not odd are even. 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.