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 85
- 85 is the smallest odd number that is not a square, not a prime, and not a prime plus a square
- 85 is the largest n for which 12+22+32+...+n2 = 1+2+3+...+m has a solution
Common Properties of 85
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.
The number n is evil if it has an even number of 1's in its binary expansion.
Guess what odious numbers are.
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.
A composite number is called a Smith number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).
In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?
A number is said to be square-free if its prime decomposition contains no repeated factors.