# 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 197

- 197 is the total sum of the digits of all two digit primes
- 197 is the smallest cyclic but not permutable prime (its cyclic rotations - 971 and 719 - are primes, but there exists a permutation - 791 or 917 - that is composite)
- 197 is the only three-digit prime Keith number. Keith numbers have the following rare property: form the sequence with the digits of n. If the total number of digits is n then later terms are sums of t previous terms. Then number n belongs to the sequence
- 197 is the smallest multidigit number which is a sum of repdigits of its digits (197 = 111 + 9 + 77)

## Common Properties of 197

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 *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.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.