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 105

105 is the smallest number such that the 105th cyclotomic polynomial has a coefficient that differs from 0, 1, or -1

Common Properties of 105

Composite

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

Composite numbers are opposite to prime numbers.

...,
100,
102,
104,
105,
106,
108,
110,
...

Deficient

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.

...,
99,
101,
103,
105,
106,
107,
109,
...

Evil

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

Guess what odious numbers are.

...,
99,
101,
102,
105,
106,
108,
111,
...

Lucky

To build the lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, ... . The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, ... . The next number remaining is 9, so delete every ninth number, etc.

Those numbers were lucky they weren't crossed out.

...,
87,
93,
99,
105,
111,
115,
127,
...

Odd

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.

...,
99,
101,
103,
105,
107,
109,
111,
...

Square-free

A number is said to be square-free if its prime decomposition contains no repeated factors.

...,
101,
102,
103,
105,
106,
107,
109,
...

Triangular

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.

...,
66,
78,
91,
105,
120,
136,
153,
...

Tanya Khovanova

Number Gossip

Unique Properties of 105

Common Properties of 105