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 496
- 496 is the smallest triangular number such that the sum of cubes of its digits is prime
- 496 is the largest perfect number with all different digits
- 496 is the smallest start of the first run of exactly 2 consecutive not squarefree triangular numbers
- 496 is the smallest even prime-indexed triangular number such that the next number is composite
Rare Properties of 496
The number n is perfect if the sum of all its positive divisors except itself is equal to n.
Less than perfect numbers are called deficient, too perfect -- abundant.
- 6,
- 28,
- 496,
- 8128,
- 33550336,
- 8589869056,
- ...
Common Properties of 496
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.
One can take the sum of the squares of the digits of a number. Those numbers are happy for which iterating this operation eventually leads to 1.
The number n is odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a triangle number.
Compare to square, pentagonal and tetrahedral numbers.