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 210
- 210 is the smallest area of two different primitive Pythagorean triangles (20, 21, 29 and 12, 35, 37)
- 210 is the product of the first 4 primes
- 210 is the smallest number such that the sum of the 4th powers powers of its digits equals the sum of its prime divisors: 210 = 2*3*5*7 and 24 + 14 + 04 = 2 + 3 + 5 + 7 = 17
Rare Properties of 210
The p-primorial is the product of all primes less than or equal to p. It is sometimes denoted by p#.
Compare to compositorials and factorials.
The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
Common Properties of 210
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.
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.
A number is said to be square-free if its prime decomposition contains no repeated factors.
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.
The untouchable numbers are those that are not the sum of the proper divisors of any number.