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Pair of integers related by their divisors
In mathematics, the 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
Amicable_numbers
Type of positive integer pairs
In mathematics, specifically number theory, betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors
Betrothed_numbers
Numbers whose aliquot sums form a cyclic sequence
sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The
Sociable_number
Integer of the form 3 × 2^n – 1 for non-negative n
Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers. The binary representation of the Thabit number 3·2n−1
Thabit_number
Open source middleware system for volunteer and grid computing
Search for primes such as Generalized Fermat primes, 321 primes, Sierpiński numbers, Cullen-Woodall primes, Proth prime, and Sophie Germain primes. Subprojects
Berkeley Open Infrastructure for Network Computing
Berkeley_Open_Infrastructure_for_Network_Computing
Two or more natural numbers with a common abundancy index
no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also
Friendly_number
Number used for counting
natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting
Natural_number
Natural number
(2013). Perfect And Amicable Numbers. World Scientific. p. 390. ISBN 9789811259647. Deza, Elena; Deza, Michel (2012). Figurative Numbers. World Scientific
69_(number)
Numbers obtained by adding the two previous ones
of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The initial elements
Fibonacci_sequence
relatively prime amicable numbers? Are there infinitely many pairs of amicable numbers? Are there infinitely many betrothed numbers? Are there infinitely
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Iranian mathematician
century. He gave the pair of amicable numbers 9,363,584 and 9,437,056 many years before Euler's contribution to amicable numbers. His major book is Oyoun
Muhammad_Baqir_Yazdi
Positive integer of the form (2^(2^n))+1
number or part of a pair of amicable numbers. (Luca 2000) The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca
Fermat_number
Mathematics
In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum
Amicable_triple
Branch of pure mathematics
mystical qualities to perfect and amicable numbers, and dedicated time to the study polygonal or figurate numbers. Later, Euclid devoted part of his
Number_theory
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Number divisible only by 1 and itself
natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite
Prime_number
Number equal to the sum of its proper divisors
numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable.
Perfect_number
Medieval Arab scholar (c. 830–901)
equation for determining amicable numbers. His proof of this rule is presented in the Treatise on the Derivation of the Amicable Numbers in an Easy Way. This
Thābit_ibn_Qurra
theorem and discovered the theorem by which pairs of amicable numbers can be found (i.e., two numbers such that each is the sum of the proper divisors of
Timeline_of_mathematics
Integer having a non-trivial divisor
integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. For example, the integer
Composite_number
Prime number of the form 2^n – 1
OEIS). Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined
Mersenne_prime
Number that cannot be written as an aliquot sum
divisors. Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, none of the Mersenne numbers are untouchable, since Mn = 2n
Untouchable_number
French mathematician and lawyer (1601–1665)
Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered Fermat's
Pierre_de_Fermat
Natural number
in the first pair of amicable numbers with 220. That means that the sum of the proper divisors are the same between the two numbers. 284 can be written
284_(number)
Natural number
a largely composite number. It is the sum of the smallest pair of amicable numbers: (220, 284). The group order of the fourth smallest non-cyclic simple
500_(number)
the Banu Musa brothers. Discovered a theorem that enables pairs of amicable numbers to be found.[citation needed] Later, al-Baghdadi (b. 980) developed
Timeline of science and engineering in the Muslim world
Timeline_of_science_and_engineering_in_the_Muslim_world
Learning aid for mathematics
educationalists Maria Montessori and Friedrich Fröbel had used rods to represent numbers, but it was Georges Cuisenaire who introduced the rods that were to be
Cuisenaire_rods
Eleventh letter in the Greek alphabet
ISBN 9780199262854. Deza, Elena (2023). Perfect and amicable numbers. Selected chapters of number theory : special numbers. New Jersey: World Scientific. p. 79.
Lambda
Concept in number theory
{\displaystyle p=1} , and an amicable narcissistic number is a sociable narcissistic number with p = 2 {\displaystyle p=2} . All natural numbers n {\displaystyle
Narcissistic_number
Mathematical recursive sequence
eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture) More unsolved problems
Aliquot_sequence
Product of an integer with itself
square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers). In the real number system, square numbers are non-negative
Square_number
Mathematical concept
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k
Lucky_numbers_of_Euler
Topics referred to by the same term
Italian composer Nicolò Paganini's numbers (c. 1850–?), Italian mathematician who found a pair of amicable numbers Paganini (disambiguation) This disambiguation
Nicolò_Paganini
Number that remains the same when its digits are reversed
the OEIS). Palindromic numbers receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain
Palindromic_number
Class of natural numbers with many divisors
any smaller positive integer. The first ten superior highly composite numbers and their factorization are listed. For a superior highly composite number
Superior highly composite number
Superior_highly_composite_number
Infinite integer series where the next number is the sum of the two preceding it
Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances
Lucas_number
Type of number introduced by Mike Keith
{\displaystyle k} terms, n {\displaystyle n} is part of the sequence. Keith numbers were introduced by Mike Keith in 1987. They are computationally very challenging
Keith_number
Numbers parameterizing ways to partition a set
of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers. The Stirling numbers of the second
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
2016 Greek film
me". It is a formula known by the pythagoreans regarding amicable numbers, pairs of numbers whose proper divisors sum to the other, like 220 and 284.
The_Other_Me_(2016_film)
Polyhedral number representing a tetrahedron
The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is, T e n = ∑ k = 1 n T k = ∑ k = 1 n k ( k + 1 ) 2 = ∑ k = 1 n (
Tetrahedral_number
Arithmetic function related to the divisors of an integer
S2CID 3207238, Zbl 1163.11059 Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", The American Mathematical Monthly, 74 (8): 969–973
Divisor_function
Numbers with a certain property involving recursive summation
function for p = 2 {\displaystyle p=2} . The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reginald Allenby (a British
Happy_number
Type of figurate number
h_{n}=2n^{2}-n=n(2n-1)={\frac {2n(2n-1)}{2}}.} The first few hexagonal numbers (sequence A000384 in the OEIS) are: 1, 6, 15, 28, 45, 66, 91, 120, 153
Hexagonal_number
Recursive integer sequence
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named
Catalan_number
Numbers whose sum of divisors is twice the number plus 1
numbers known: 3, 10, 136, 32896 and 2147516416 (sequence A191363 in the OEIS) Betrothed numbers relate to quasiperfect numbers like amicable numbers
Quasiperfect_number
Count of permutations by cycles
combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the unsigned Stirling numbers of the first kind count
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Sum of all proper divisors of a natural number
define s(0) = 0). Sociable numbers are numbers whose aliquot sequence is a periodic sequence. Amicable numbers are sociable numbers whose aliquot sequence
Aliquot_sum
Takagi existence theorem (number theory) Thabit ibn Qurra's theorem (amicable numbers) Thue's theorem (Diophantine equation) Thue–Siegel–Roth theorem (Diophantine
List_of_theorems
Persian mathematician (1265–1318)
number theory is on amicable numbers. In Tadhkira al-ahbab fi bayan al-tahabb ("Memorandum for friends on the proof of amicability") introduced a major
Kamāl_al-Dīn_al-Fārisī
Figurate number
triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally
Pentagonal_number
Integer having only small prime factors
the natural numbers. 5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are also called humble numbers, and sometimes
Smooth_number
Number of stacked spheres in a pyramid
study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming
Square_pyramidal_number
Type of figurate number
regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks
Polygonal_number
Type of composite number with an even number of digits
number with an even number of digits, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors
Vampire_number
Numbers that evenly divide powers of 60
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors
Regular_number
Mathematical sequences in combinatorics
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in
Stirling_number
Number raised to the third power
stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number
Cube_(algebra)
Integers occurring in the coefficients of the Taylor series of 1/cosh t
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e
Euler_numbers
Sequence in number theory
and a amicable Dudeney root is a sociable Dudeney root with k = 2 {\displaystyle k=2} . Sociable Dudeney numbers and amicable Dudeney numbers are the
Dudeney_number
Book by Denis Guedj
topics covered in the book include primes and factors; irrational and amicable numbers; the discoveries of Pythagoras, Archimedes and Euclid; and the problems
The_Parrot's_Theorem
Positive integer that is the product of three distinct prime numbers
prime numbers, there are also infinitely many sphenic numbers. A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In
Sphenic_number
Size of a geometric arrangement of points
triangular numbers, polygonal numbers, tetrahedral numbers, and pyramidal numbers, and subsequent mathematicians have included other classes of these numbers including
Figurate_number
Egyptian mathematician (1194–1252)
Pietro Cataldi Brentjes, Sonja (1988). "The First Perfect Numbers and Three Types of Amicable Numbers in a Manuscript on Elementary Number Theory by Ibn Fallûs"
Ibn_Fallus
Count of the possible partitions of a set
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th
Bell_number
Product of two prime numbers
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
Semiprime
Wiki-based programming chrestomathy
Bottles of Beer" (song) Abbreviations Ackermann function Amicable numbers Anagrams Bernoulli numbers Bitwise operations Cholesky decomposition Combinations
Rosetta_Code
Integer divisible by sum of its digits
digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Because being a harshad number is determined
Harshad_number
2012-01-13. "BOINC Stats — Albert@home". BOINC. Retrieved 2012-02-17. "Amicable Numbers - Detailed stats". Retrieved 2023-03-03. "First commit ·
List of volunteer computing projects
List_of_volunteer_computing_projects
Type of figurate number constructed by combining heptagons
{\displaystyle H_{n}={\frac {5n^{2}-3n}{2}}} . The first few heptagonal numbers are: 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540
Heptagonal_number
Integer whose multiples are digit rotations
excluded: single digits, e.g.: 5 repeated digits, e.g.: 555 repeated cyclic numbers, e.g.: 142857142857 If leading zeros are not permitted on numerals, then
Cyclic_number
Number used to approximate the square root of 2
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational
Pell_number
Number that is more than the sum of its proper divisors
However, he applied this classification only to the even numbers. Almost perfect number Amicable number Sociable number Superabundant number Prielipp (1970)
Deficient_number
Mathematical sequence
In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They
Lah_number
Base-dependent property of integers
number is a sociable Kaprekar number with k = 1 {\displaystyle k=1} , and a amicable Kaprekar number is a sociable Kaprekar number with k = 2 {\displaystyle
Kaprekar_number
Characterization of even perfect numbers
S2CID 13607242 Euler, Leonhard (1849), "De numeris amicibilibus" [On amicable numbers], Commentationes arithmeticae (in Latin), vol. 2, pp. 627–636. Originally
Euclid–Euler_theorem
Type of composite integer
numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed. Smith numbers were
Smith_number
Iterative algorithm on numbers
and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10: 9981 – 1899 =
Kaprekar's_routine
Odd number with specific properties
× 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd
Sierpiński_number
Type of natural number
Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28
Hyperperfect_number
Notable events in the history of algebra
theorem, and discovered the theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of
Timeline_of_algebra
Type of positive integer
ones cancel out the effect of their signs. Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up
Amenable_number
Number that is abundant but not semiperfect
mathematics Are there any odd weird numbers? More unsolved problems in mathematics Infinitely many weird numbers exist. For example, 70p is weird for
Weird_number
901 – Thabit Ibn Qurra, discovered a theorem which enables pairs of amicable numbers to be found 847 to 871 – Emirate of Bari 850 to 934 – Abu Zayd al-Balkhi
Timeline of Middle Eastern history
Timeline_of_Middle_Eastern_history
Medal (2014) Muhammad Baqir Yazdi (17th century), found the pair of amicable numbers 9,363,584 and 9,437,056 Nasir Khusraw (1004–1088), scientist, Ismaili
List of Iranian mathematicians
List_of_Iranian_mathematicians
Composite number in number theory
absolute test of primality. The Carmichael numbers form the subset K1 of the Knödel numbers. The Carmichael numbers were named after the American mathematician
Carmichael_number
Centered figurate number
checkers is played on. The numbers are also called centered dodecagonal numbers because star numbers are centered polygonal numbers with a twelve-sided shape
Star_number
Mathematical sequence
In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanisław Ulam, who introduced it in 1964. The standard Ulam
Ulam_number
Integer filtered out using a sieve similar to that of Eratosthenes
eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers). The term
Lucky_number
Ten raised to an integer power
as 1En in E notation. See order of magnitude and orders of magnitude (numbers) for named powers of ten. There are two conventions for naming positive
Power_of_10
Triangular array of natural numbers
In combinatorics, the Narayana numbers N ( n , k ) , n ∈ N + , 1 ≤ k ≤ n {\displaystyle \operatorname {N} (n,k),n\in \mathbb {N} ^{+},1\leq k\leq n}
Narayana_number
Number equal to the sum of all or some of its divisors
all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in the OEIS)
Semiperfect_number
Class of natural numbers
abundant numbers are superabundant numbers. 0-super abundant numbers are highly composite numbers. For example, generalized 2-super abundant numbers are 1
Superabundant_number
Class of binary number
non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and for
Evil_number
(that cannot be turned over) 1183 = pentagonal pyramidal number 1184 = amicable number with 1210 1185 = number of partitions of 45 into pairwise relatively
1000_(number)
Number that represents a hexagon with a dot in the center
centered hexagonal numbers: Centered hexagonal numbers should not be confused with cornered hexagonal numbers, which are figurate numbers in which the associated
Centered_hexagonal_number
Number whose square ends in the same digits
{Z} _{b}} , the ring of b {\displaystyle b} -adic integers, automorphic numbers are used to find the numerical representations of the fixed points of f
Automorphic_number
Type of Poulet number
number. The super-Poulet numbers below 10,000 are (sequence A050217 in the OEIS): It is relatively easy to get super-Poulet numbers with 3 distinct prime
Super-Poulet_number
Number with a half-integer abundancy index
hemiperfect numbers of abundancy k/2 for k ≤ 13 (sequence A088912 in the OEIS): The current best known upper bounds for the smallest numbers of abundancy
Hemiperfect_number
Thabit ibn Qurra gives a theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of
Timeline_of_number_theory
Number, product of consecutive integers
The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term
Pronic_number
AMICABLE NUMBERS
AMICABLE NUMBERS
Girl/Female
Christian & English(British/American/Australian)
Amiable
Male
Dutch
, amiable.
Boy/Male
Tamil
Amiable
Biblical
lovely, amiable
Girl/Female
Tamil
Amiable, Praiseworthy
Boy/Male
Arabic, Muslim
Sociable; Amicable; Friendly
Girl/Female
Hindu
Amiable, Praiseworthy
Biblical
amiable; beloved
Male
Danish
, amiable.
Boy/Male
British, English, Welsh
Affection; Amiable
Male
French
French name derived from Latin amabilis, AMABLE means "lovable."
Girl/Female
British, Dutch, English, German, Swedish
Lovable; Amiable
Girl/Female
Biblical
Amiable, beloved.
Girl/Female
Muslim
Amicable, Friendly
Boy/Male
Greek
Amiable.
Girl/Female
Arabic, Muslim
Amicable; Friendly
Male
Danish
, amiable.
Boy/Male
Hindu
Amiable
Girl/Female
Indian, Tamil
Admirable Language; Admirable Word
Girl/Female
Arabic, Christian
Admirable
AMICABLE NUMBERS
AMICABLE NUMBERS
Girl/Female
Hindu
Shy
Boy/Male
American, Gujarati, Hindu, Indian, Punjabi, Sikh
A Flower
Female
French
French form of English Agnes, INÈS means "chaste; holy."
Boy/Male
Afghan, Arabic, Kashmiri, Muslim
Servant of the Most Compassionate
Boy/Male
Arabic, Muslim
Gift of the Merciful Allah
Boy/Male
English
Lives at the Castle
Boy/Male
Indian
Might of the faith
Girl/Female
Biblical
A window, grief.
Boy/Male
Indian, Punjabi, Sikh
Teachings of Guru
Boy/Male
Muslim
Scholar. LittTrateur.
AMICABLE NUMBERS
AMICABLE NUMBERS
AMICABLE NUMBERS
AMICABLE NUMBERS
AMICABLE NUMBERS
a.
Friendly; kindly; sweet; gracious; as, an amiable temper or mood; amiable ideas.
adv.
In an amiable manner.
a.
Wonderful; also, admirable.
adv.
In an amenable manner.
a.
Pleasing; amiable.
a.
Not amovable or removable.
a.
Worthy of imitation; as, imitable character or qualities.
adv.
In an amicable manner.
a.
Possessing sweetness of disposition; having sweetness of temper, kind-heartedness, etc., which causes one to be liked; as, an amiable woman.
n.
The quality of being amicable; friendliness; amicableness.
n.
The quality of being amiable; amiability.
a.
Appropriate to, or implying, friendship; befitting friends; amicable.
a.
Wonderful; admirable.
a.
Not mixable.
a.
Liable to be brought to account or punishment; answerable; responsible; accountable; as, amenable to law.
n.
The quality of being amicable; amicability.
a.
Such as can be mined; as, minable earth.
n.
A breaking up of amicable relations; rupture.
a.
Friendly; proceeding from, or exhibiting, friendliness; after the manner of friends; peaceable; as, an amicable disposition, or arrangement.
a.
Not amiable; morose; ill-natured; repulsive.