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

• ...,
• 2500,
• 2501,
• 2507,
• 2511,
• 2527,
• 2528,
• 2530,
• ...

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

Composite numbers are opposite to prime numbers.

• ...,
• 2508,
• 2509,
• 2510,
• 2511,
• 2512,
• 2513,
• 2514,
• ...

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.

• ...,
• 2507,
• 2509,
• 2510,
• 2511,
• 2512,
• 2513,
• 2515,
• ...

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

Guess what odious numbers are.

• ...,
• 2505,
• 2506,
• 2508,
• 2511,
• 2513,
• 2514,
• 2516,
• ...

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.

• ...,
• 2484,
• 2485,
• 2487,
• 2511,
• 2517,
• 2518,
• 2524,
• ...

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.

• ...,
• 2491,
• 2493,
• 2505,
• 2511,
• 2523,
• 2527,
• 2545,
• ...

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.

• ...,
• 2505,
• 2507,
• 2509,
• 2511,
• 2513,
• 2515,
• 2517,
• ...

The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.

• ...,
• 2481,
• 2484,
• 2486,
• 2511,
• 2513,
• 2525,
• 2550,
• ...