Properties

Is a natural number (positive integer)

Is an integer

Is a rational number

Is a real number

Is a complex number

An even number is an integer that is divisible by 2, i.e., there exists an integer k such that n = 2k.

An odd number is an integer that is NOT divisible by 2, i.e., there exists an integer k such that n = 2k + 1.

Euler's totient function φ(n) counts the positive integers up to n that are relatively prime to n (i.e., gcd(k, n) = 1 for 1 ≤ k ≤ n).

The Möbius function μ(n) is defined as:

The radical of a positive integer n, denoted rad(n), is the product of all distinct prime factors of n.

The big omega function Ω(n) counts the prime factors of n with multiplicity.

The little omega function ω(n) counts the distinct prime factors of n.

The digit count of a positive integer n is the number of digits in its decimal representation.

The digit sum of a non-negative integer is the sum of all its digits in its decimal representation.

The digital root of a positive integer is the single-digit value obtained by repeatedly summing the digits until only one digit remains.

A palindromic number is a number that remains the same when its digits are reversed.

A repdigit (repeated digit) is a positive integer composed entirely of the same digit.

An automorphic number (or circular number) is a number whose square ends with the number itself.

A Kaprekar number is a positive integer n such that when n² is split into two parts (left and right), the sum of those parts equals n.

The multiplicative persistence of a number n is the number of times you must multiply its digits together until you reach a single digit.

The divisor count (or divisor function τ(n), also written d(n)) counts the number of positive divisors of n, including 1 and n itself.

The divisor sum σ(n) is the sum of all positive divisors of n, including 1 and n itself.

The proper divisor sum (or aliquot sum) s(n) is the sum of all positive divisors of n excluding n itself.

An abundant number (or excessive number) is a positive integer n for which the sum of its proper divisors exceeds n itself.

A perfect number is a positive integer that equals the sum of its proper divisors (all divisors excluding itself).

A deficient number is a positive integer n for which the sum of its proper divisors is less than n.

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer.

A weird number is a natural number that is: 1. Abundant: Sum of proper divisors > n 2. Not semiperfect: Cannot equal any subset sum of its proper divisors

An untouchable number is a positive integer that is NOT the sum of proper divisors of any number.

A primitive abundant number is an abundant number none of whose proper divisors are abundant.

A Fibonacci number is any number that appears in the Fibonacci sequence, where each number is the sum of the two preceding ones.

A triangular number counts objects that can form an equilateral triangle. The nth triangular number is the sum of the first n positive integers:

A perfect square (or square number) is an integer that can be expressed as the square of another integer.

A perfect cube (or cube number) is an integer that can be expressed as the cube of another integer.

A pentagonal number counts objects that can form a regular pentagon pattern. The nth pentagonal number is:

A hexagonal number counts objects that can form a regular hexagon pattern. The nth hexagonal number is:

A number m is a Lucas number if it appears in the Lucas sequence, defined by:

A tetrahedral number Teₙ is a figurate number that represents a tetrahedron (triangular pyramid). It equals the sum of the first n triangular numbers:

A number m is a Pell number if it appears in the Pell sequence defined by:

A number m is a Tribonacci number if it appears in the Tribonacci sequence:

A pronic number (or oblong/heteromecic number) is the product of two consecutive integers:

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

A composite number is a positive integer greater than 1 that has at least one positive divisor other than 1 and itself.

A semiprime (also called biprime or 2-almost prime) is a natural number that is the product of exactly two prime numbers (not necessarily distinct).

A squarefree number is a positive integer that is not divisible by any perfect square other than 1.

A prime power is a positive integer that can be expressed as p^k where p is a prime and k ≥ 1.

The prime factorization of a positive integer is its unique representation as a product of prime powers.

A twin prime is a prime number that is part of a twin prime pair — two primes that differ by exactly 2.

A Mersenne prime is a prime number of the form 2^p - 1, where p itself must be prime.

A Sophie Germain prime is a prime p such that 2p + 1 is also prime.

A safe prime is a prime q of the form q = 2p + 1, where p is also prime.

A powerful number (or squareful number) is a positive integer n such that for every prime p dividing n, p² also divides n.

An Achilles number is a positive integer that is: 1. Powerful — every prime factor appears with exponent ≥ 2 2. Not a perfect power — cannot be written as m^k for any m, k ≥ 2

The prime index or prime-counting function π(n) gives the number of primes less than or equal to n.

A perfect power is a positive integer n that can be expressed as m^k where m ≥ 1 and k ≥ 2.

A B-smooth number (or B-friable number) is a positive integer whose prime factors are all at most B.

A Carmichael number is a composite number n that satisfies Fermat's Little Theorem for ALL bases coprime to n:

A happy number is a positive integer that eventually reaches 1 when replaced repeatedly by the sum of the squares of its digits.

A Harshad number (or Niven number) is a positive integer that is divisible by the sum of its digits.

A narcissistic number (also called Armstrong number or pluperfect digital invariant) is a number that equals the sum of its digits each raised to the power of the number of digits.

The Collatz steps (or total stopping time) of a positive integer n is the number of iterations needed to reach 1 using the Collatz function:

A Smith number is a composite number where the sum of its digits equals the sum of the digits of its prime factorization (with multiplicity).

A vampire number is a composite natural number with an even number of digits, that can be factored into two integers (called "fangs") each with half as many digits, where the digits of the original number can be rearranged to form the fangs.

A number n is a factorial number if n = k! for some non-negative integer k.

A number m is a Catalan number if m = Cₙ for some non-negative integer n, where:

A number m is a Bell number if m = Bₙ for some non-negative integer n, where Bₙ counts the number of ways to partition a set of n elements into non-empty subsets.

A number m is a Motzkin number if m = Mₙ for some n, where Mₙ counts: - Lattice paths from (0,0) to (n,0) using up, down, and flat steps - Non-intersecting chord diagrams on n points (with unpaired points allowed)

Lucky numbers are generated by a sieve similar to Eratosthenes, but using position-based elimination:

A number m is a primorial if m = p