




They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.
First ten: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54
There are 2487 abundant numbers below 10,000.
It all started with perfect numbers which are amicable with themselves. Those numbers adopted social virtues and qualities; for the parts of each of them have the power to generate the other. See also sociable numbers.
First ten: 220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368
There are 10 amicable numbers below 10,000.
First ten: 157, 192, 218, 220, 222, 224, 226, 243, 245, 247
There are 6485 apocalyptic powers below 10,000.
First ten: 25, 95, 119, 143, 417, 445, 565, 608, 650, 652
There are 89 aspiring numbers below 10,000.
It is curious, how for a kdigit automorphic number n there is another automorphic number  10^{k} + 1  n. For this to work with n=1, you have to treat 1 as a zerodigit number.
First ten: 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376.
There are 8 automorphic numbers below 10,000.
Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.
First ten: 2, 4, 8, 15, 26, 42, 64, 93, 130, 176.
There are 38 cake numbers below 10,000.
Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.
First ten: 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341.
There are 7 Carmichael numbers below 10,000.
There are many ways Catalan numbers can be interpreted; there are some cool pictures here and the Wikipedia article is very good.
First ten: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796.
There are 9 Catalan numbers below 10,000.
Composite numbers are opposite to prime numbers.
First ten: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
There are 8769 composite numbers below 10,000.
Compositorial numbers are factorials divided by primorials.
First ten: 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000.
There are 4 compositorials below 10,000.
First ten: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
There are 21 cube numbers below 10,000.
Compare with perfect and abundant numbers.
First ten: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11.
There are 7508 deficient numbers below 10,000.
Numbers that are not even are odd. Compare with another pair  evil and odious numbers.
First ten: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
There are 4999 even numbers below 10,000.
Guess what odious numbers are.
First ten: 3, 5, 6, 9, 10, 12, 15, 17, 18, 20.
There are 4999 evil numbers below 10,000.
The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.
First ten: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800.
There are 7 factorials below 10,000.
Fibonacci numbers are very common in nature. For example, a pinapple has 8 spirals if you count one way, and 13 if you count the other way.
First ten: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
There are 19 different Fibonacci numbers below 10,000.
They are named Google numbers because of the unusual hiring ad that Google put up.
First ten: 2, 71, 271, 4523, 74713, 904523, 2718281, 72407663, 360287471, 7427466391.
There are 4 Google numbers below 10,000.
First ten: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44.
There are 1441 happy numbers below 10,000.
They are named hungry numbers because they try to eat as much "pi" as possible.
First ten: 5, 17, 74, 144, 144, 2003, 2003, 37929, 82810, 161449.
There are 5 different hungry numbers below 10,000.
Unlike the situation with cake, everybody gets the toppings.
First ten: 2, 4, 7, 11, 16, 22, 29, 37, 46, 56.
There are 140 lazy caterer numbers below 10,000.
Those numbers were lucky they weren't crossed out.
First ten: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33.
There are 1118 lucky numbers below 10,000.
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.
First ten: 3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911.
There are 6 Mersenne numbers below 10,000.
The drive to find prime numbers among Mersenne numbers supplies humanity with the largest known prime numbers.
First ten: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111.
There are 5 Mersenne primes below 10,000.
First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153.
There are 16 narcissistic numbers below 10,000.
Numbers that are not odd are even. Compare with another pair  evil and odious numbers.
First ten: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
There are 5000 odd numbers below 10,000.
Guess what evil numbers are.
First ten: 1, 2, 4, 7, 8, 11, 13, 14, 16, 19.
There are 5000 odious numbers below 10,000.
First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11.
There are 198 palindromic numbers below 10,000.
In base 2 Mersenne primes are palindromic primes.
First ten: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191.
There are 20 palindromic primes below 10,000.
Pentagonal numbers are to pentagons what triangular numbers are to triangles and square numbers are to squares.
First ten: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.
There are 81 pentagonal numbers below 10,000.
Less than perfect numbers are called deficient, too perfect  abundant.
First ten: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216.
There are 4 perfect numbers below 10,000.
How much power? They all can be written as a^{2} b^{3}.
First ten: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49.
There are 184 powerful numbers below 10,000.
First ten: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
There are 14 powers of 2 below 10,000.
First ten: 1, 2, 4, 6, 8, 12, 16, 18, 20, 24.
There are 1455 practical numbers below 10,000.
Prime numbers are opposite to composite numbers.
First ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
There are 1229 primes below 10,000.
Compare to compositorials and factorials.
First ten: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230.
There are 5 primorials below 10,000.
They are twice triangular numbers.
First ten: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110.
There are 99 pronic numbers below 10,000.
The term repunit comes from combining "repeated" and "unit".
First ten: 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111.
There are 4 repunits below 10,000.
In 1984, when Albert Wilansky called his brotherinlaw, named Smith, he noticed that the phone number posesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brotherinlaw :) ?
First ten: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265.
There are 376 Smith numbers below 10,000.
The 2cycles are the amicable pairs and the 1cycles are the perfect numbers. For some numbers it is very difficult to compute the aliquot sequence. The smallest number whose sequence has not been completely computed is 276.
First ten (known): 12496, 14264, 14288, 14316, 14536, 15472, 17716, 19116, 19916, 22744.
There are no sociable numbers below 10,000.
First ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
There are 99 squares below 10,000.
First ten: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14.
There are 6083 squarefree numbers below 10,000.
This is the space generalization of triangular and square numbers.
First ten: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220.
There are 83 tetrahedral numbers below 10,000.
Compare to square, pentagonal and tetrahedral numbers.
First ten: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
There are 140 triangular numbers below 10,000.
First ten: 3, 5, 7, 11, 13, 17, 19, 29, 31, 41.
There are 409 twin primes below 10,000.
First ten: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18.
There are 827 Ulam numbers below 10,000.
This property is significant starting from 3digit numbers, so we will not consider numbers below 100.
First ten: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191.
There are 180 undulating numbers below 10,000.
First ten: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188.
There are 1212 untouchable numbers below 10,000.
First ten: 126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530.
There are 15 vampire numbers below 10,000.
First ten: 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792.
There are 7 weird numbers below 10,000.