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

• ...,
• 1713,
• 1715,
• 1722,
• 1723,
• 1739,
• 1743,
• 1744,
• ...

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.

• ...,
• 1718,
• 1719,
• 1721,
• 1723,
• 1724,
• 1725,
• 1726,
• ...

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

Guess what odious numbers are.

• ...,
• 1716,
• 1719,
• 1720,
• 1723,
• 1725,
• 1726,
• 1728,
• ...

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.

• ...,
• 1701,
• 1705,
• 1711,
• 1723,
• 1731,
• 1737,
• 1749,
• ...

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.

• ...,
• 1717,
• 1719,
• 1721,
• 1723,
• 1725,
• 1727,
• 1729,
• ...

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.

• ...,
• 1699,
• 1709,
• 1721,
• 1723,
• 1733,
• 1741,
• 1747,
• ...

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

• ...,
• 1718,
• 1721,
• 1722,
• 1723,
• 1726,
• 1727,
• 1729,
• ...

A prime number is called a twin prime if there exists another prime number differing from it by 2.

• ...,
• 1697,
• 1699,
• 1721,
• 1723,
• 1787,
• 1789,
• 1871,
• ...