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 * 33 * 7 = 23 + 33 + 73)
- 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.