Amicable number

Amicable numbers are two different numbers so related that the sum of the proper divisors of the one is equal to the other, one being considered as a proper divisor but not the number itself. Such a pair is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220. Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties.

A pair of amicable numbers constitutes an aliquot sequence of period 2.

A general formula by which these numbers could be derived was invented circa 850 by Thabit ibn Qurra (826-901): if :"p" = 3 × 2"n" − 1 − 1, :"q" = 3 × 2"n" − 1,:"r" = 9 × 22"n" − 1 − 1, where "n" > 1 is an integer and "p", "q", and "r" are prime numbers, then 2"npq" and 2"nr" are a pair of amicable numbers. This formula gives the amicable pair (220, 284), as well as the pair (17296, 18416) and the pair (9363584, 9437056). The pair (6232, 6368) are amicable, but they cannot be derived from this formula. In fact, this formula produces amicable numbers for "n" = 2, 4, and 7, but for no other values below 20000.

In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists. Also, every known pair shares at least one common factor. It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.

Amicable numbers have been studied by Al Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980-1037), Al-Farisi (1260-1320), René Descartes (1596-1650), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others. Thabit's formula was generalized by Euler. Prior to Euler only three pairs of amicable numbers had been found. Because Euler found 59 more amicable numbers, the work of Eastern mathematicians in this area is largely forgotten.

The pair (9363584; 9437056) has often been attributed to Descartes, but it was actually first discovered by Muhammad Baqir Yazdi in Iran. [cite journal | last = Costello |first = Patrick | title = New Amicable Pairs Of Type (2; 2) And Type (3; 2) | journal = MATHEMATICS OF COMPUTATION | volume = 72 Number 241 | pages = 489–497 | publisher = American Mathematical Society | date = 2002-05-01 | url = http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf | accessdate = 2007-04-19 ]

The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) OEIS|id=A063990

If a number equals the sum of "its own" proper divisors, it is called a perfect number.

The following Python language code allows you to check if two numbers are Amicable:

# Definition of the functiondef amicable_numbers(x,y): #only two different numbers may be amicable if x = y: return False # Sum all values i in [1,x) where i divides x sum_x = sum(i for i in xrange(1, x) if x % i = 0) sum_y = sum(i for i in xrange(1, y) if y % i = 0) return (sum_x = y) and (sum_y = x)

# Program bodyn_1=int(raw_input('Enter nº 1: '))n_2=int(raw_input('Enter nº 2: '))

if amicable_numbers(n_1,n_2): print 'Amicable! :)'else: print 'Not Amicable :('

And the following PseudoCode finds all the Amicable Numbers between two numbers

Procedure Find Amicable Pairs Enter Starting Number Enter Last Number For all the numbers between the Starting Number and Last Number and call this FirstNumber Call the Function to add all of the Proper Divisors of the FirstNumber and call this SumOfAllProperDivisorsOfFirstNumber Call the Function to add all of the Proper Divisors again this time using SumOfAllProperDivisorsOfFirstNumber and call this SumOfAllProperDivisorsOfSecondNumber If SumOfAllProperDivisorsOfFirstNumber is equal to SumOfAllProperDivisorsOfSecondNumber then You found a pair End if End For LoopEnd of Procedure

Function Add All Of The Proper Divisors of A Number (call this ANumber) Set the initial Running Total to 0 For all the numbers between 1 and half of ANumber and call this CurrentLoopNumber If you divide ANumber with CurrentLoopNumber and the remainder is zero then Add the result to the Running Total End If End For Loop Return the Running TotalEnd of Function

References

* Wells, D. (1987). "The Penguin Dictionary of Curious and Interesting Numbers" (pp. 145 - 147). London: Penguin Group.

External links

* [http://amicable.homepage.dk/knwnc2.htm All known amicable numbers]
* [http://ftp.cwi.nl/CWIreports/MAS/MAS-R0307.pdf A good 2003 survey on current status of Amicable number mathematics.]


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Look at other dictionaries:

  • amicable number — noun Either of a pair of amicable numbers …   Wiktionary

  • amicable number — Math. either of a pair of positive integers in which each member is equal to the sum of the submultiples of the other, as 220 and 284. * * * …   Universalium

  • amicable number — noun : either of two numbers each of which is equal to the sum of all the submultiples of the other * * * Math. either of a pair of positive integers in which each member is equal to the sum of the submultiples of the other, as 220 and 284 …   Useful english dictionary

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  • Number theory — A Lehmer sieve an analog computer once used for finding primes and solving simple diophantine equations. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers (the… …   Wikipedia

  • amicable numbers — noun A pair of numbers having the property that the sum of the divisors of each, excluding itself, is equal to the other number. The factors of 220, excluding 220 itself, are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and sum to 284; the factors… …   Wiktionary

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  • 60000 (number) — Number number = 60000 range = 10000 100000 cardinal = 60000 ordinal = th ordinal text = sixty thousandth factorization = 2^5 cdot 3 cdot 5^4 bin = 1110101001100000 oct = 165140 hex = EA6060,000 (sixty thousand) is the number that comes after… …   Wikipedia

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  • 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… …   Wikipedia

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