The composite integer n is a Carmichael number if b^n-1 = 1 (mod n) for every integer b which is relatively prime with n.

Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.

• 561,
• 1105,
• 1729,
• 2465,
• 2821,
• ...

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

• ...,
• 1092,
• 1102,
• 1103,
• 1105,
• 1128,
• 1133,
• 1135,
• ...

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

Composite numbers are opposite to prime numbers.

• ...,
• 1101,
• 1102,
• 1104,
• 1105,
• 1106,
• 1107,
• 1108,
• ...

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.

• ...,
• 1101,
• 1102,
• 1103,
• 1105,
• 1106,
• 1107,
• 1108,
• ...

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

Guess what odious numbers are.

• ...,
• 1098,
• 1100,
• 1103,
• 1105,
• 1106,
• 1108,
• 1111,
• ...

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.

• ...,
• 1093,
• 1095,
• 1101,
• 1105,
• 1107,
• 1117,
• 1123,
• ...

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.

• ...,
• 1099,
• 1101,
• 1103,
• 1105,
• 1107,
• 1109,
• 1111,
• ...

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

• ...,
• 1101,
• 1102,
• 1103,
• 1105,
• 1106,
• 1109,
• 1110,
• ...