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 72
- 72 is the smallest Achilles number, a number that is powerful but not a perfect power
- 72 is the smallest number that can be expressed as the difference of the squares of consecutive primes in two distinct ways, (72 = 192 - 172 = 112 - 72)
- 72 is the only number of the form pq · qp that lies between twin primes, where p and q are primes
Rare Properties of 72
The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
Common Properties of 72
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.
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.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.