Definition: The number n is abundant if the sum of all its positive divisors except itself is more than n.
First ten: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54
amicable
Definition: The number n is amicable if it belongs to an amicable pair. Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.
First ten: 220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368
Definition:The number n is called an apocalyptic power if 2n contains the consecutive digits 666 (in decimal).
First ten: 157, 192, 218, 220, 222, 224, 226, 243, 245, 247
aspiring
Definition:The number n is called an aspiring number if its aliquot sequence terminates in a perfect number, and it is not a perfect number itself.
First ten: 25, 95, 119, 143, 417, 445, 565, 608, 650, 652
automorphic(curious)
Definition:The number n is called an automorphic number if (the decimal expansion of) n2 ends with n. These numbers are also called curious.
First ten: 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376.
cake
Definition:The n-th cake number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.
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
Definition: The composite integer n is a Carmichael number if bn-1 = 1 (mod n) for every integer b which is relatively prime with n.
First ten: 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341.
Catalan
Definition: The n-th Catalan number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).
First ten: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796.
composite
Definition: A positive integer greater than 1 that is not prime is called composite.
First ten: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
compositorial
Definition: The n-th compositorial is the product of the first n composite numbers.
First ten: 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000.
cube
Definition: The number n is a cube if it is the cube of an integer.
First ten: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
deficient
Definition: The number n is deficient if the sum of all its positive divisors except itself is less than n.
First ten: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11.
even
Definition: A number is even if it is divisible by 2.
First ten: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
evil
Definition: The number n is evil if it has an even number of 1’s in its binary expansion.
First ten: 3, 5, 6, 9, 10, 12, 15, 17, 18, 20.
factorial
Definition: The n-th factorial is the product of the first n natural numbers.
First ten: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800.
Fibonacci
Definition: 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 preceeding ones.
First ten: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Definition: The n-th Google number is the first n-digit prime found in the decimal expansion of e.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.
googol
Definition: 1 followed by 100 zeros
First and Last number: 1×10^100
Definition: 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.
First ten: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44.
hungry
Definition: The k-th hungry number is the smallest number n such that 2^n contains the first k digits of the decimal expansion of pi.
First ten: 5, 17, 74, 144, 144, 2003, 2003, 37929, 82810, 161449.
lazy caterer
Definition: 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.
First ten: 2, 4, 7, 11, 16, 22, 29, 37, 46, 56.
lucky
Definition: 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.
First ten: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33.
Mersenne
Definition: A number of the form 2p - 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.
First ten: 3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911.
Mersenne prime
Definition: 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.
First ten: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111.
narcissistic
Definition: 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.
First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153.
odd
Definition: 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.
First ten: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
odious
Definition: The number n is odious if it has an odd number of 1’s in its binary expansion.Guess what evil numbers are.
First ten: 1, 2, 4, 7, 8, 11, 13, 14, 16, 19.
palindrome
Definition: A palindrome is a number that reads the same forward or backward.
First ten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11.
palindromic prime
Definition: A palindromic prime is a prime which is a palindrome.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
Definition: Pentagonal numbers are of the form n(3n - 1)/2.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.
perfect
Definition: The number n is perfect if the sum of all its positive divisors except itself is equal to n.Less than perfect numbers are called deficient, too perfect — abundant.
First ten: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216.
powerful
Definition: 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.
First ten: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49.
power of 2
Definition: A number is a power of 2 if it is 2 to some power.
First ten: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
practical
Definition: The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
First ten: 1, 2, 4, 6, 8, 12, 16, 18, 20, 24.
prime
Definition: 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.
First ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
primorial
Definition: The p-primorial is the product of all primes less than or equal to p. It is sometimes denoted by p#.Compare to compositorials and factorials.
First ten: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230.
pronic(heteromecic)
Definition: The number is called pronic if it is the product of two consecutive numbers.
First ten: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110.
repunit
Definition: A repunit is an integer in which every digit is one.
First ten: 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111.
Smith (joke)
Definition: 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).
First ten: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265.
sociable
Definition: An aliquot sequence is formed by taking an integer, adding all of its divisors other than itself, and then repeating this process with the sum. The numbers for which this process returns to the starting point after more than two steps are called sociable numbers. The 2-cycles are the amicable pairs and the 1-cycles 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.
square
Definition: The number n is a square if it is the square of an integer.
First ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
square-free
Definition: A number is said to be square-free if its prime decomposition contains no repeated factors.
First ten: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14.
tetrahedral(pyramidal)
Definition: A tetrahedral number is the number of balls you can put in a triangular pyramid.
First ten: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220.
triangular
Definition: 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.
First ten: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
twin
Definition: A prime number is called a twin prime if there exists another prime number differing from it by 2.
First ten: 3, 5, 7, 11, 13, 17, 19, 29, 31, 41.
Ulam
Definition: The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.
First ten: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18.
undulating
Definition: Undulating numbers are numbers of the form abababab… in base 10.
First ten: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191.
untouchable
Definition: The untouchable numbers are those that are not the sum of the proper divisors of any number.
First ten: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188.
vampire
Definition: The number n is called a vampire number if there exists a factorization of n using n’s digits.
First ten: 126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530.
weird
Definition: The number n is called a weird number if it is abundant but is not the sum of any subset of its proper factors.
First ten: 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792.
Sources:
http://www.tanyakhovanova.com/Numbers/numbers.html#lazy%20caterer
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