Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

Unique Properties of 21

21 is the smallest number of distinct integer-sided squares needed to tile a square
21 is the number of spots on a standard cubical die (1+2+3+4+5+6)

Rare Properties of 21

Fibonacci numbers are numbers that form the Fibonacci sequence. The Fibonacci sequence is defined as starting with 1, 1 and then each next term is the sum of the two preceding ones.

Fibonacci numbers are very common in nature. For example, a pineapple has 8 spirals if you count one way, and 13 if you count the other way.

...,
5,
8,
13,
21,
34,
55,
89,
...

Common Properties of 21

Composite

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

Composite numbers are opposite to prime numbers.

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16,
18,
20,
21,
22,
24,
25,
...

Deficient

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.

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16,
17,
19,
21,
22,
23,
25,
...

Lucky

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.

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9,
13,
15,
21,
25,
31,
33,
...

Odd

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.

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15,
17,
19,
21,
23,
25,
27,
...

Odious

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

Guess what evil numbers are.

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14,
16,
19,
21,
22,
25,
26,
...

Square-free

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

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15,
17,
19,
21,
22,
23,
26,
...

Triangular

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a triangle number.

Compare to square, pentagonal and tetrahedral numbers.

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6,
10,
15,
21,
28,
36,
45,
...

Tanya Khovanova

Number Gossip

Unique Properties of 21

Rare Properties of 21

Common Properties of 21