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 1111
- 1111 is the sum of the digits of the first 100 primes
- 1111 is the smallest number n for which there are exactly ten k such that n equals k plus the reverse of k
- 1111 is the smallest number such that its square contains 1234 as a substring (11112 = 1234321)
- 1111 is the smallest composite number having the same digits as their prime factors (with multiplicity), excluding zero digits (1111 = 11*101)
- 1111 is the smallest strobogrammatic composite numbes whose prime factors are strobogrammatic as well
Rare Properties of 1111
A repunit is an integer in which every digit is one.
The term repunit comes from combining "repeated" and "unit".
- 1,
- 11,
- 111,
- 1111,
- 11111,
- 111111,
- 1111111,
- ...
Common Properties of 1111
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 palindrome is a number that reads the same forward or backward.
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.
Undulating numbers are numbers of the form abababab... in base 10.
This property is significant starting from 3-digit numbers, so we will not consider numbers below 100.