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

• ...,
• 6364,
• 6365,
• 6366,
• 6367,
• 6368,
• 6369,
• 6370,
• ...

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.

• ...,
• 6363,
• 6364,
• 6365,
• 6367,
• 6368,
• 6369,
• 6371,
• ...

One can take the sum of the squares of the digits of a number. Those numbers are happy for which iterating this operation eventually leads to 1.

• ...,
• 6354,
• 6361,
• 6364,
• 6367,
• 6370,
• 6373,
• 6376,
• ...

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.

• ...,
• 6351,
• 6355,
• 6363,
• 6367,
• 6369,
• 6373,
• 6379,
• ...

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.

• ...,
• 6361,
• 6363,
• 6365,
• 6367,
• 6369,
• 6371,
• 6373,
• ...

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

Guess what evil numbers are.

• ...,
• 6361,
• 6362,
• 6364,
• 6367,
• 6368,
• 6371,
• 6373,
• ...

A prime is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

• ...,
• 6353,
• 6359,
• 6361,
• 6367,
• 6373,
• 6379,
• 6389,
• ...

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

• ...,
• 6362,
• 6365,
• 6366,
• 6367,
• 6369,
• 6371,
• 6373,
• ...