The number n is called a vampire number if there exists a factorization of n using n's digits (counting multiplicity).

• ...,
• 1503,
• 1530,
• 1827,
• 2187,
• 3159,
• 3784,
• 6880,
• ...

The number n is called an apocalyptic power if 2ⁿ contains the consecutive digits 666 (in decimal).

• ...,
• 2180,
• 2185,
• 2186,
• 2187,
• 2189,
• 2195,
• 2200,
• ...

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

Composite numbers are opposite to prime numbers.

• ...,
• 2184,
• 2185,
• 2186,
• 2187,
• 2188,
• 2189,
• 2190,
• ...

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.

• ...,
• 2183,
• 2185,
• 2186,
• 2187,
• 2188,
• 2189,
• 2191,
• ...

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.

• ...,
• 2133,
• 2163,
• 2173,
• 2187,
• 2209,
• 2211,
• 2215,
• ...

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.

• ...,
• 2181,
• 2183,
• 2185,
• 2187,
• 2189,
• 2191,
• 2193,
• ...

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

Guess what evil numbers are.

• ...,
• 2180,
• 2183,
• 2184,
• 2187,
• 2189,
• 2190,
• 2192,
• ...

An integer n is powerful if for every prime p dividing n, p² also divides n.

How much power? They all can be written as a² b³.

• ...,
• 2025,
• 2048,
• 2116,
• 2187,
• 2197,
• 2209,
• 2304,
• ...