The number n is a cube if it is the cube of an integer.

• ...,
• 216,
• 343,
• 512,
• 729,
• 1000,
• 1331,
• 1728,
• ...

The number n is a square if it is the square of an integer.

• ...,
• 576,
• 625,
• 676,
• 729,
• 784,
• 841,
• 900,
• ...

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

Composite numbers are opposite to prime numbers.

• ...,
• 725,
• 726,
• 728,
• 729,
• 730,
• 731,
• 732,
• ...

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.

• ...,
• 724,
• 725,
• 727,
• 729,
• 730,
• 731,
• 733,
• ...

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

Guess what odious numbers are.

• ...,
• 723,
• 725,
• 726,
• 729,
• 730,
• 732,
• 735,
• ...

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.

• ...,
• 717,
• 723,
• 727,
• 729,
• 735,
• 739,
• 741,
• ...

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.

• ...,
• 723,
• 725,
• 727,
• 729,
• 731,
• 733,
• 735,
• ...

An integer n is powerful if for every prime p dividing n, p² also divides n.

How much power? They all can be written as a² b³.

• ...,
• 648,
• 675,
• 676,
• 729,
• 784,
• 800,
• 841,
• ...

A composite number is called a Smith number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?

• ...,
• 690,
• 706,
• 728,
• 729,
• 762,
• 778,
• 825,
• ...