# 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 313

- 313 is the smallest multidigit palindromic prime that is both a left- and right-truncatable prime
- 313 is the only 3-digit palindromic prime which is palindromic in base 2 too
- 313 is the smallest palindromic prime such that the sum of its digits is prime and the number of prime digits is prime
- 313 is the smallest number such that the average of the digits of its square is 8
- 313 is the smallest prime whose smallest positive primitive root is 10
- 313 is the smallest number n such that the sum of the divisors of 2n
^{2} is a square

## Common Properties of 313

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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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.