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This is a list of recreational number theory topics (see
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
,
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
). Listing here is not
pejorative A pejorative or slur is a word or grammatical form expressing a negative or a disrespectful connotation, a low opinion, or a lack of respect toward someone or something. It is also used to express criticism, hostility, or disregard. Sometimes, a ...
: many famous topics in number theory have origins in challenging problems posed purely for their own sake. See
list of number theory topics {{Short description, none This is a list of number theory topics, by Wikipedia page. See also: *List of recreational number theory topics *Topics in cryptography Divisibility *Composite number **Highly composite number *Even and odd numbers **Par ...
for pages dealing with aspects of number theory with more consolidated theories.


Number sequences

{{cleanup list, date=May 2013 *
Integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
*
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
** Golden mean base **
Fibonacci coding In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains n ...
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Lucas sequence In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this rec ...
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Padovan sequence In number theory, the Padovan sequence is the sequence of integers ''P''(''n'') defined. by the initial values :P(0)=P(1)=P(2)=1, and the recurrence relation :P(n)=P(n-2)+P(n-3). The first few values of ''P''(''n'') are :1, 1, 1, 2, 2, 3, 4, 5 ...
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Figurate numbers The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygo ...
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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 are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples T ...
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Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
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Square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usu ...
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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. The ...
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Hexagonal number A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...
***
Heptagonal number A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula :H_n=\frac. The first few heptagonal numbers are: : 0, 1, 7, 18, 34, 55, 81, 112 ...
***
Octagonal number An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 34 ...
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Nonagonal number A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the constr ...
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Decagonal number A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal number ...
**Centered polygonal 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 cen ...
***
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. The centered pentagonal number for ''n'' is given by th ...
*** Centered hexagonal 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. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
**Pyramidal number *** Triangular pyramidal number *** Square pyramidal number ***
Pentagonal pyramidal number A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. ...
*** Hexagonal pyramidal number ***
Heptagonal pyramidal number A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. ...
**
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. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahed ...
**
Star number A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on. The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The ...
*
Perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...
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Quasiperfect number In mathematics, a quasiperfect number is a natural number ''n'' for which the sum of all its divisors (the divisor function ''σ''(''n'')) is equal to 2''n'' + 1. Equivalently, ''n'' is the sum of its non-trivial divisors (that is, its divisors excl ...
**
Almost perfect number In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' â ...
**
Multiply perfect number Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
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Hyperperfect number In mathematics, a ''k''-hyperperfect number is a natural number ''n'' for which the equality ''n'' = 1 + ''k''(''σ''(''n'') − ''n'' − 1) holds, where ''σ''(''n'') is the divisor function (i.e., the sum of all positive divisors of ''n ...
**
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 **
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 (a divisor ''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n''/''d'' share no common factors). Some perfect ...
**
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 (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divis ...
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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 (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. ...
*
Amicable number Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive d ...
*
Sociable number In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and ...
*
Abundant number In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. Th ...
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Deficient number In number theory, a deficient number or defective number is a number ''n'' for which the sum of divisors of ''n'' is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than ''n''. For ex ...
* Amenable number *
Aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Defi ...
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Super-Poulet number A super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor ''d'' divides :2''d'' − 2. For example, 341 is a super-Poulet number: it has positive divisors and we have: :(211 - 2) / 11 = 2046 / 11 = 186 :(231 - 2) ...
*
Lucky number In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remain ...
*
Powerful number A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 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'' = ''a ...
*
Primeval number In recreational number theory, 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 (in base 10) is larger than the number of primes obtainable in the same w ...
* Palindromic number * Triangular square number *
Harmonic divisor number In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are: : 1, 6, 2 ...
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Sphenic number In number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definit ...
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Smith number In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the given number base. In the case of numbers that are not square-f ...
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Double Mersenne number In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and Li ...
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Zeisel number A Zeisel number, named after Helmut Zeisel, is a square-free integer ''k'' with at least three prime factors which fall into the pattern :p_x = ap_ + b where ''a'' and ''b'' are some integer constants and ''x'' is the index number of each prime f ...
* Heteromecic number * Niven numbers *
Superparticular number In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers. More particularly, the ratio takes the form: :\frac = 1 + \frac where is a positive integer. Thu ...
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Highly composite number __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...
* Highly totient number * Practical number *
Juggler sequence In number theory, a juggler sequence is an integer sequence that starts with a positive integer ''a''0, with each subsequent term in the sequence defined by the recurrence relation: a_= \begin \left \lfloor a_k^ \right \rfloor, & \text a_k \text ...
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Look-and-say sequence In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: : 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... . To generate a member of the sequence from the previous member, read off t ...


Digits

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Polydivisible number In mathematics a polydivisible number (or magic number) is a number in a given number base 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 mul ...
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Automorphic number In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natura ...
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Armstrong number In number theory, a narcissistic number 1 F_ : \mathbb \rightarrow \mathbb to be the following: : F_(n) = \sum_^ d_i^k. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, and : d_i = \frac is the value of each d ...
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Self number In number theory, a self number or Devlali number in a given number base b is a natural number that cannot be written as the sum of any other natural number n and the individual digits of n. 20 is a self number (in base 10), because no such combina ...
* Harshad number *
Keith number In number theory, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number n in a given number base b with k digits such that when a sequence is created such that the first k terms are the k digits of n and ...
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Kaprekar number In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. The numbers are n ...
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Digit sum In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. We define the digit ...
* Persistence of a number * Perfect digital invariant **
Happy number In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy number because ...
* Perfect digit-to-digit invariant * Factorion *
Emirp An emirp (''prime'' spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term ''reversible prime'' is used to mean the same as e ...
* Palindromic prime *
Home prime In number theory, the home prime HP(''n'') of an integer ''n'' greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The ''m''th intermediate stage in the process o ...
*
Normal number In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to b ...
** Stoneham number ** Champernowne constant ** Absolutely normal number *
Repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreat ...
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Repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Example ...


Prime and related sequences

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Semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
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Almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if Ω(''n'') = ''k'', where Ω(''n'') is the total number of primes in the prime fact ...
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Unique prime The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737. Like all rational numbers, the reciprocals of primes have repeating decimal represen ...
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Factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! +&n ...
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Permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study the ...
* Palindromic prime *
Cuban prime A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers ''x'' and ''y''. First series This is the first of these equations: :p = \frac,\ x = ...
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Lucky prime In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remain ...


Magic squares, etc.

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Ulam spiral The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's ''Mathematical Games'' column in ''Scientific American'' a short time late ...
* Magic star * Magic square **
Frénicle standard form A magic square is in the Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions hold: # the element at position ,1(top left corner) is the smallest of the four corner elements; and # the element at position ...
** Prime reciprocal magic square ** Trimagic square ** Multimagic square **
Panmagic square A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the squar ...
** Satanic square **
Most-perfect magic square A most-perfect magic square of order ''n'' is a magic square containing the numbers 1 to ''n''2 with two additional properties: # Each 2 × 2 subsquare sums to 2''s'', where ''s'' = ''n''2 + 1. # All pairs of ...
** Geometric magic square ** Conway's Lux method for magic squares *
Magic cube In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an ''n'' × ''n'' × ''n'' pattern such that the sums of the numbers on each row, on each c ...
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Perfect magic cube Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * ''Perfect'' (1985 film), a romantic drama * ''Perfect'' (2018 film), a science ...
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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 Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineeri ...
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Bimagic 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 powers for 1 ≤ ''k'' ≤ ''P''. cubes are called bimagic, cubes are called trimagic, and cubes t ...
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Trimagic 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 powers for 1 ≤ ''k'' ≤ ''P''. cubes are called bimagic, cubes are called trimagic, and cubes t ...
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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 powers for 1 ≤ ''k'' ≤ ''P''. cubes are called bimagic, cubes are called trimagic, and cubes t ...
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Magic hypercube In mathematics, a magic hypercube is the ''k''-dimensional generalization of magic squares and magic cubes, that is, an ''n'' × ''n'' × ''n'' × ... × ''n'' array of integers such that the sums of the numbers on each pillar (along any axis) a ...
* Magic constant *
Squaring the square Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the sq ...
Recreational number theory *