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 248
- 248 is the smallest 3-digit numbers with distinct digits greater than one, such that it is divisible by all of its digits
- 248 is the smallest number n, greater than 1, for which the arithmetic, geometric, and harmonic means of φ(n) and σ(n) are all integers (φ(248) = 120 and σ(248) = 480; there arithmetic, geometric and harmonic means are integers: 300, 240, and 192)
- 248 is the dimension of E8, the largest exceptional simple Lie algebra (submitted by Qiaochu Yuan)
Common Properties of 248
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 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 odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
The untouchable numbers are those that are not the sum of the proper divisors of any number.