Amicable number

Article on other languages:

del.icio.us del.icio.us
Digg Digg
Furl Furl
Reddit Reddit
Rojo Rojo
Add to OnlyWire
Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization

Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers 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 × 2n − 1 − 1,
q = 3 × 2n − 1,
r = 9 × 22n − 1 − 1,

where n > 1 is an integer and p, q, and r are prime numbers, then 2npq and 2nr 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.[1]

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

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 function
def 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 range(1, x) if x % i == 0)
    sum_y = sum(i for i in range(1, y) if y % i == 0)
    return (sum_x == y) and (sum_y == x)
 
# Program body
n_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 Loop
End 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 Total
End of Function

References

Look up amicable in
Wiktionary, the free dictionary.

This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.

  • Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (pp. 145 - 147). London: Penguin Group.
  1. ^ Costello, Patrick (2002-05-01). "New Amicable Pairs Of Type (2; 2) And Type (3; 2)". MATHEMATICS OF COMPUTATION (American Mathematical Society) 72 Number 241: 489–497, http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf. Retrieved on 19 April 2007. 

External links

Retrieved from "http://en.wikipedia.org/wiki/Amicable_number"

This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.