Perfect number number whose divisors (except itself) add up to itself. For example,. 28 is a perfect number. Its proper divisors are {1, 2, 4, 7, 14} and 1 +2 +4 +7 +14 = 28. Can you find the smallest perfect number?
Deficient number number whose proper divisors add up to less than itself. For example, 16 is a deficient number as sum of its proper divisors {1, 2, 4, 8} = 15.
Abundant number number whose proper divisors add up to more than the number itself. For example, 18 is a abundant number as 1+2 +3 +6 +9 = 21. Can you find the smallest abundant number?
Square number number that can be formed by counting the number of objects used in making a square

1 is a square number. 
4 is a square number.  
9 is a square number. 
Can you find the next three square numbers?
Triangular number number that can be formed by counting the number of objects used in making a triangle.
1 is a triangular number  
3 is a triangular number  
6 is a triangular number 
Can you find the next three triangular numbers?
Happy number number for which the sum of the squares of the digits eventually equals 1.For example, 203 is happy because 2² +0² +3² = 13; 1² +3² = 10; 1² +0² = 1
Can you think of another happy number? An unhappy number?
Narcissistic number like self centered persons, the narcissistic numbers are integers equal to an expression using the same digits. For example, 36 is narcissistic as 36 = 3! × 6
Sometimes, narcissistic number is defined as a number equal to the sum of its digits raised to a certain power (mostly the number of digits in the number).
371 is narcissistic as 371 = 3³ +7³ +1³
Can you think of another narcissistic number?
Palindromic number number having same digits forward and backward e.g. 121, 102201, 8523258 etc.
Prime number numbers which have on other divisors than 1 and itself . 2, 3, 5, 7, 11, 13, 17,.......... etc. are prime numbers. They have fascinated mathematicians since ages.
Other kind of numbers Transcendental numbers (Liouville numbers), Vulgar numbers (Irrational numbers), Cyclic numbers, Sociable numbers, Fibonacci numbers, Amicable numbers, Coprime numbers, Figurate numbers, Weird numbers etc.
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Some special numbers:
0 : Zero Discovered by Indians. Read more at The story of zero
2 : The only even Prime
2 : Black money 
: (sade sati ) denotes misery and suffering
007 : Spy (of James Bond variety)
10 : (Fraud )
13 : Lucky in some countries, unlucky in some. Many hotels in Europe don't have room number 13.
16 : Sweet sixteen
420 : All Indians are aware of this section on cheating in Criminal Code.
666 : The number of the beast.
786 : Lucky number for Muslims. Amitabh Bachhan wore batch number 786 in "Coolie".
1729 : Number associated with Ramanujan.