A number of the form 2^p - 1 is called a Mersenne number if p is prime.

It was believed many years ago, that all Mersenne numbers are prime. This is not so, thus there is a separate entry for Mersenne prime numbers.

• 3,
• 7,
• 31,
• 127,
• 2047,
• ...

A Mersenne number which is also prime is called a Mersenne prime.

The drive to find prime numbers among Mersenne numbers supplies humanity with the largest known prime numbers.

• 3,
• 7,
• 31,
• 127,
• 8191,
• ...

A k-digit number n is called narcissistic if it is equal to the sum of k-th powers of its digits. They are also called Plus Perfect numbers.

• ...,
• 4,
• 5,
• 6,
• 7,
• 8,
• 9,
• 153,
• ...

A palindromic prime is a prime which is a palindrome.

In base 2 Mersenne primes are palindromic primes.

• 2,
• 3,
• 5,
• 7,
• 11,
• 101,
• 131,
• ...

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.

• ...,
• 3,
• 4,
• 5,
• 7,
• 8,
• 9,
• 10,
• ...

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.

• 1,
• 7,
• 10,
• 13,
• 19,
• ...

The n-th lazy caterer number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

• 2,
• 4,
• 7,
• 11,
• 16,
• 22,
• ...

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.

• 1,
• 3,
• 7,
• 9,
• 13,
• 15,
• ...

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.

• 1,
• 3,
• 5,
• 7,
• 9,
• 11,
• 13,
• ...

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

Guess what evil numbers are.

• 1,
• 2,
• 4,
• 7,
• 8,
• 11,
• 13,
• ...

A palindrome is a number that reads the same forward or backward.

• ...,
• 4,
• 5,
• 6,
• 7,
• 8,
• 9,
• 11,
• ...

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.

• 2,
• 3,
• 5,
• 7,
• 11,
• 13,
• 17,
• ...

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

• ...,
• 3,
• 5,
• 6,
• 7,
• 10,
• 11,
• 13,
• ...

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

• 3,
• 5,
• 7,
• 11,
• 13,
• 17,
• ...