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 181
- 181 is the smallest multi-digit palindromic prime with an odd number of digits, such that increasing the middle digit by 1 produces another prime
- 181 is the start of the first occurrence of two successive primes (181, 191) using no prime digits
- 181 starts the first occurrence of two consecutive primes which are both multidigit palindromes: 181, 191
- 181 is the smallest palindromic prime with digit sum 10
- 181 is the smallest number n such that the binary complement of n is greater than the binary complement of n2
- 181 is the smallest number that have more zeros in its binary representation compared to its square
Common Properties of 181
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.
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.
The number n is odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
A palindrome is a number that reads the same forward or backward.
A prime is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.
Prime numbers are opposite to composite numbers.
A number is said to be square-free if its prime decomposition contains no repeated factors.
A prime number is called a twin prime if there exists another prime number differing from it by 2.
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.