Number Gossip Properties
- apocalyptic power
- lazy caterer
- Mersenne prime
- palindromic prime
- power of 2
Definition: The number n is abundant if the sum of all its positive divisors except itself is more than n.
There are 2487 abundant numbers below 10,000.
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.
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.
There are 10 amicable numbers below 10,000.
Definition: The number n is called an apocalyptic power if 2n contains the consecutive digits 666 (in decimal).
There are 6485 apocalyptic powers below 10,000.
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.
There are 89 known aspiring numbers below 10,000.
Definition: The number n is called an automorphic number if (the decimal expansion of) n2 ends with n. These numbers are also called curious.
It is curious, how for a k-digit automorphic number n there is another automorphic number -- 10k + 1 - n. For this to work with n=1, you have to treat 1 as a zero-digit number.
There are 8 automorphic numbers below 10,000.
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.
There are 39 cake numbers below 10,000.
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.
Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.
There are 7 Carmichael numbers below 10,000.
Definition: The n-th Catalan number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).
There are 9 Catalan numbers below 10,000.
Definition: A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
There are 8769 composite numbers below 10,000.
Definition: The n-th compositorial is the product of the first n composite numbers.
There are 4 compositorials below 10,000.
Definition: The number n is a cube if it is the cube of an integer.
There are 21 cube numbers below 10,000.
Definition: The number n is deficient if the sum of all its positive divisors except itself is less than n.
There are 7508 deficient numbers below 10,000.
Definition: A number is even if it is divisible by 2.
There are 4999 even numbers below 10,000.
Definition: The number n is evil if it has an even number of 1's in its binary expansion.
Guess what odious numbers are.
There are 4999 evil numbers below 10,000.
Definition: The n-th factorial is the product of the first n natural numbers.
The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.
There are 7 factorials below 10,000.
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 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.
There are 19 different Fibonacci numbers below 10,000.
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.
There are 4 Google numbers below 10,000.
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.
There are 1441 happy numbers below 10,000.
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.
They are named hungry numbers because they try to eat as much "pi" as possible.
There are 7 hungry numbers below 10,000.
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.
There are 140 lazy caterer numbers below 10,000.
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.
There are 1118 lucky numbers below 10,000.
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.
There are 6 Mersenne numbers below 10,000.
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.
There are 5 Mersenne primes below 10,000.
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.
There are 16 narcissistic numbers below 10,000.
Definition: A number is odd if it is not divisible by 2.
There are 5000 odd numbers below 10,000.
Definition: The number n is odious if it has an odd number of 1's in its binary expansion.
Guess what evil numbers are.
There are 5000 odious numbers below 10,000.
Definition: A palindrome is a number that reads the same forward or backward.
There are 198 palindromic numbers below 10,000.
In base 2 Mersenne primes are palindromic primes.
There are 20 palindromic primes below 10,000.
Definition: Pentagonal numbers are of the form n(3n - 1)/2.
There are 81 pentagonal numbers below 10,000.
Definition: The number n is perfect if the sum of all its positive divisors except itself is equal to n.
There are 4 perfect numbers below 10,000.
Power of 2
Definition: A number is a power of 2 if it is 2 to some power.
There are 14 powers of 2 below 10,000.
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.
There are 184 powerful numbers below 10,000.
Definition: The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
There are 1455 practical numbers below 10,000.
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.
There are 1229 primes below 10,000.
Definition: The p-primorial is the product of all primes less than or equal to p. It is sometimes denoted by p#.
There are 5 primorials below 10,000.
Definition: The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
There are 99 pronic numbers below 10,000.
Definition: A repunit is an integer in which every digit is one.
The term repunit comes from combining "repeated" and "unit".
There are 4 repunits below 10,000.
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).
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 :-) ?
There are 376 Smith numbers below 10,000.
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
There are no known sociable numbers below 10,000.
Definition: The number n is a square if it is the square of an integer.
There are 99 squares below 10,000.
Definition: A number is said to be square-free if its prime decomposition contains no repeated factors.
There are 6083 square-free numbers below 10,000.
Definition: A tetrahedral number is the number of balls you can put in a triangular pyramid.
There are 38 tetrahedral numbers below 10,000.
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.
There are 140 triangular numbers below 10,000.
Definition: A prime number is called a twin prime if there exists another prime number differing from it by 2.
There are 409 twin primes below 10,000.
Definition: The next Ulam number is uniquely the sum of two earlier distinct Ulam numbers.
There are 827 Ulam numbers below 10,000.
Definition: Undulating numbers are numbers of the form abababab... in base 10.
This property is significant starting from 3-digit numbers, so we will not consider numbers below 100.
There are 180 undulating numbers below 10,000.
Definition: The untouchable numbers are those that are not the sum of the proper divisors of any number.
There are 1212 untouchable numbers below 10,000.
Definition: The number n is called a vampire number if there exists a factorization of n using n's digits.
There are 15 vampire numbers below 10,000.