Pentagonal numbers are of the form n(3n - 1)/2.

Pentagonal numbers are to pentagons what triangular numbers are to triangles and square numbers are to squares.

• ...,
• 12,
• 22,
• 35,
• 51,
• 70,
• 92,
• 117,
• ...

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

Composite numbers are opposite to prime numbers.

• ...,
• 48,
• 49,
• 50,
• 51,
• 52,
• 54,
• 55,
• ...

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.

• ...,
• 47,
• 49,
• 50,
• 51,
• 52,
• 53,
• 55,
• ...

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

Guess what odious numbers are.

• ...,
• 45,
• 46,
• 48,
• 51,
• 53,
• 54,
• 57,
• ...

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.

• ...,
• 37,
• 43,
• 49,
• 51,
• 63,
• 67,
• 69,
• ...

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.

• ...,
• 45,
• 47,
• 49,
• 51,
• 53,
• 55,
• 57,
• ...

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

• ...,
• 43,
• 46,
• 47,
• 51,
• 53,
• 55,
• 57,
• ...