Number sequences

• Integer sequence
  Integer sequence
  In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
  
  
• Fibonacci sequence
  • Golden mean base
  • Fibonacci coding
    Fibonacci coding
    In mathematics, Fibonacci coding is a universal code which encodes positive integers into binary code words. Each code word ends with "11" and contains no other instances of "11" before the end.-Definition:...
    
    
• Lucas sequence
  Lucas sequence
  In mathematics, the Lucas sequences Un and Vn are certain integer sequences that satisfy the recurrence relationwhere P and Q are fixed integers...
  
  
• Padovan sequence
  Padovan sequence
  The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...
  
  
• Figurate numbers
• Polygonal number
  Polygonal number
  In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots were thought of as alphas . These are one type of 2-dimensional figurate numbers.- Definition and examples :...
  
  
  • Triangular number
    Triangular number
    A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
    
    
  • Square number
    Square number
    In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
    
    
• Pentagonal number
  Pentagonal number
  A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical...
  
  
• Hexagonal number
  Hexagonal number
  A hexagonal number is a figurate number. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.The formula for the nth hexagonal number...
  
  
• Heptagonal number
• Octagonal number
• Nonagonal number
• Decagonal number
  Decagonal number
  A decagonal number is a figurate number that represents a decagon. The n-th decagonal number is given by the formulaThe first few decagonal numbers are:...
  
  
• Centered square number
  Centered square number
  In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a...
  
  
• Centered pentagonal number
  Centered pentagonal number
  A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers...
  
  
• Centered hexagonal number
  Centered hexagonal number
  A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
  
  
• Tetrahedral number
  Tetrahedral number
  A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron...
  
  
• Pyramidal number
  • Triangular pyramidal number
  • Square pyramidal number
    Square pyramidal number
    In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base...
    
    
  • Pentagonal pyramidal number
    Pentagonal pyramidal number
    A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base. The nth pentagonal pyramidal number is equal to the sum of the first n pentagonal numbers....
    
    
  • Heptagonal pyramidal number
    Heptagonal pyramidal number
    A heptagonal pyramidal number is the sum of the first few heptagonal numbers. The heptagonal number for n can be calculated by adding up the heptagonal numbers for 1 to n, or by using the formula n/6....
    
    
• Octahedral number
  Octahedral number
  In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres...
  
  
• Star number
  Star number
  A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...
  
  
• Perfect number
  Perfect number
  In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
  
  
  • Quasiperfect number
    Quasiperfect number
    In mathematics, a quasiperfect number is a theoretical natural number n for which the sum of all its divisors is equal to 2n + 1...
    
    
  • Almost perfect number
    Almost perfect number
    In mathematics, an almost perfect number is a natural number n such that the sum of all divisors of n is equal to 2n - 1, the sum of all proper divisors of n, s = σ - n, then being equal to n - 1...
    
    
  • Multiply perfect number
    Multiply perfect number
    In mathematics, a multiply perfect number is a generalization of a perfect number....
    
    
  • Hyperperfect number
    Hyperperfect number
    In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k holds, where σ is the divisor function . A hyperperfect number is a k-hyperperfect number for some integer k...
    
    
  • Semiperfect number
    Semiperfect number
    In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number....
    
    
  • Primitive semiperfect number
    Primitive semiperfect number
    In number theory, a primitive semiperfect number is a semiperfect number that has no semiperfect proper divisor....
    
    
  • Unitary perfect number
    Unitary perfect number
    A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.Thus, 60 is a unitary perfect...
    
    
  • Weird number
    Weird number
    In number theory, a weird number is a natural number that is abundant but not semiperfect.In other words, the sum of the proper divisors of the number is greater than the number, but no subset of those divisors sums to the number itself.- Examples :The smallest weird number is 70...
    
    
• Amicable number
  Amicable number
  Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
  
  
• Sociable number
  Sociable number
  Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself...
  
  
• Abundant number
• Deficient number
  Deficient number
  In number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which...
  
  
• Amenable number
• Aliquot sequence
  Aliquot sequence
  In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way:For example, the...
  
  
• Super-Poulet number
• Lucky number
  Lucky number
  In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.Begin with a list of integers starting with 1:...
  
  
• Happy number
  Happy number
  A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 , or it loops endlessly in a cycle which does not include 1...
  
  
• Powerful number
  Powerful number
  A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...
  
  
• Primeval number
  Primeval number
  In mathematics, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits is larger than the number of primes obtainable in the same way for any smaller natural number...
  
  
• Palindromic number
  Palindromic number
  A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters...
  
  
• Automorphic number
  Automorphic number
  In mathematics an automorphic number is a number whose square "ends" in the same digits as number itself. For example, 52 = 25, 762 = 5776, and 8906252 = 793212890625, so 5, 76 and 890625 are all automorphic numbers.The sequence of automorphic numbers begins 1, 5, 6, 25, 76, 376, 625, 9376, .....
  
  
• Triangular square number
• Smith numbers
• Polydivisible number
  Polydivisible number
  In mathematics a polydivisible number is a number with digits abcde... that has the following properties :# Its first digit a is not 0.# The number formed by its first two digits ab is a multiple of 2....
  
  
• Harmonic divisor number
  Harmonic divisor number
  In mathematics, a harmonic divisor number, or Ore number , is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are...
  
  
• Sphenic number
  Sphenic number
  In number theory, a sphenic number is a positive integer which is the product of three distinct prime numbers.Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors; e.g. 60 = 22 × 3 × 5 has exactly 3 prime factors, but is not sphenic.All...
  
  
• Keith number
• Kaprekar number
  Kaprekar number
  In mathematics, a Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45. The Kaprekar numbers are...
  
  
• Smith number
  Smith number
  A Smith number is a composite number for which, in a given base , the sum of its digits is equal to the sum of the digits in its prime factorization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 =...
  
  
• Harshad number
  Harshad number
  A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit + , meaning joy-giver. The Niven...
  
  
• Double Mersenne number
• Zeisel number
• Heteromecic number
• Niven numbers
• Superparticular number
• Untouchable number
  Untouchable number
  An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer ....
  
  
• Self number
  Self number
  A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is,...
  
  
• Highly totient number
  Highly totient number
  A highly totient number k is an integer that has more solutions to the equation φ = k, where φ is Euler's totient function, than any integer below it. The first few highly totient numbers are...
  
  
• Practical number
  Practical number
  In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n...
  
  
• Armstrong number
• Juggler sequence

Digits

• Digit sum
  Digit sum
  In mathematics, the digit sum of a given integer is the sum of all its digits,...
  
  
• Persistence of a number
  Persistence of a number
  In mathematics, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point, i.e...
  
  
• Emirp
  Emirp
  An emirp is a prime number that results in a different prime when its digits are reversed. This definition excludes the related palindromic primes. Emirps are also called reversible primes....
  
  
• Palindromic prime
  Palindromic prime
  A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
  
  
• Home prime
  Home prime
  In number theory, the home prime HP of an integer n greater than 1 is the prime obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP is designated HPn...
  
  
• Normal number
  Normal number
  In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...
  
  
  • Stoneham number
    Stoneham number
    In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Stoneham . For coprime numbers b, c > 1, the Stoneham number αb,c is defined as...
    
    
  • Champernowne constant
    Champernowne constant
    In mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after mathematician D. G...
    
    
  • Absolutely normal number
• Repunit
  Repunit
  In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
  
  
• Repdigit
  Repdigit
  In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system....
  
  

Prime and related sequences

• Semiprime
  Semiprime
  In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....
  
  
• Almost prime
• Unique prime
  Unique prime
  In number theory, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q...
  
  
• Factorial prime
  Factorial prime
  A factorial prime is a prime number that is one less or one more than a factorial . The first few factorial primes are:n! − 1 is prime for :n! + 1 is prime for :...
  
  
• Permutable prime
  Permutable prime
  A permutable prime is a prime number, which, in a given base, can have its digits' positions switched through any permutation and still spell a prime number. H. E...
  
  
• Palindromic prime
  Palindromic prime
  A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
  
  
• Cuban prime
• Lucky prime

Magic squares, etc.

• Ulam spiral
  Ulam spiral
  The Ulam spiral, or prime spiral is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes...
  
  
• Magic star
• Magic square
  Magic square
  In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
  
  
  • Frénicle standard form
    Frénicle standard form
    A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:# the element at position [1,1] is the smallest of the four corner elements; and...
    
    
  • Prime reciprocal magic square
    Prime reciprocal magic square
    A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.Consider a number divided into one, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0·3333... However, the remainders of 1/7 repeat over six, or...
    
    
  • Trimagic square
  • Multimagic square
    Multimagic square
    In mathematics, a P-multimagic square is a magic square that remains magic even if all its numbers are replaced by their kth power for 1 ≤ k ≤ P...
    
    
  • Panmagic square
    Panmagic square
    A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e...
    
    
  • Satanic square
  • Most-perfect magic square
    Most-perfect magic square
    A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties:# Each 2×2 subsquare sums to 2s, where s = n2 + 1....
    
    
  • Conway's Lux method for magic squares
    Conway's LUX method for magic squares
    Conway's LUX method for magic squares is an algorithm by John Horton Conway for creating magic squares of order 4n+2, where n is a natural number.-Method:Start by creating a -by- square array consisting of...
    
    
• Magic cube
  Magic cube
  In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...
  
  
  • Perfect magic cube
    Perfect magic cube
    In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant....
    
    
  • Semiperfect magic cube
    Semiperfect magic cube
    In mathematics, a semiperfect magic cube is a magic cube that is not a perfect magic cube, i.e., a magic cube for which the cross section diagonals do not necessarily sum up to the cube's magic constant....
    
    
  • Bimagic cube
  • Trimagic cube
  • Multimagic cube
    Multimagic cube
    In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic...
    
    
• Magic tesseract
  Magic tesseract
  In mathematics, a magic tesseract is the 4-dimensional counterpart of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar as well as the main space diagonals is equal to a single number,...
  
  
• Perfect magic tesseract
• Semiperfect magic tesseract
• Magic hypercube
  Magic hypercube
  In mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers arranged in an n × n × n × .....
  
  
• Magic constant
  Magic constant
  The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
  
  
• Squaring the square
  Squaring the square
  Squaring the square is the problem of tiling an integral square using only other integral squares. The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set...