# 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 154

- The diagonals of a regular nonagon divide it into 154 regions
- 154 is the smallest starting index of a string of exactly 3 consecutive equal digits in decimal expansion of Pi
- 154 = (1*5*4)*(1+5+4) - pi((1*5*4)*(1+5+4)), where pi(n) is the number of primes less or equal to n. Note that 154 is the smallest multidigit number with this property

## Common Properties of 154

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.

A number is *even* if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

The number n is *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

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 said to be *square-free* if its prime decomposition contains no repeated factors.