# 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 191

- 191 is the smallest palindromic prime whose sum of digits is a multidigit palindromic prime
- By adding up the values of the common US coins, one obtains 191 cents (silver dollar + half dollar + quarter + dime + nickel + penny)
- 191 is the smallest palindromic prime such that the next palindromic prime has the same non-zero digit's product (the next palindromic prime is 313)
- 191 is the larger of the first occurrence of two successive primes using no prime digits (181, 191)
- 191 is the smallest multidigit palindromic prime that yields a palindrome when multiplied by the following prime
- 191 is the largest three-digit mountain prime

## Common Properties of 191

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 n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

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.