# 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 378

- 378 is the smallest 3-digit number in which the sum of digits is equal to the sum of prime factors (378 = 2 * 3 * 3 * 3 * 7 and 2 + 3 + 3 + 3 + 7 = 3 + 7 + 8)
- 378 is the smallest Smith triangular number
- 378 is the smallest number that is equal to the sum of cubes of its prime factors (378 = 2 * 3
^{3} * 7 = 2^{3} + 3^{3} + 7^{3})
- 378 is the smallest Smith abundant number
- 378 is the smallest triangular number that is a product of three distinct non-trivial triangular numbers (378 = 3 * 6 * 21)

## Rare Properties of 378

The n-th *cake* number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.

Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.

## Common Properties of 378

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime 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 number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.

A composite number is called a *Smith* number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.