# Dictionary Definition

amicable adj : characterized by friendship and
good will [ant: hostile]

# User Contributed Dictionary

## English

### Pronunciation

- /ˈæmɪcəbl/

### Adjective

- Showing friendliness or goodwill.
- They hoped to reach an amicable agreement.
- He was an amicable fellow with an easy smile.

#### Derived terms

#### Translations

showing friendliness or goodwill

- Finnish: sovinnollinen, sopuisa, ystävällinen
- German: einvernehmlich (agreement), freundlich (person), freundschaftlich (agreement, discussion, gesture), gütlich (settlement), liebenswürdig (person)
- Portuguese: amigável

# Extensive Definition

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 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,

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.

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

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

## 0: sum_x+=i for k in range(1,y): if y%k

0: sum_y+=k return sum_x## y and sum_y

x- Program body

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

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

## External links

amicable in Belarusian (Tarashkevitsa):
Прыхільныя лікі

amicable in Breton: Niver karantezus

amicable in Catalan: Nombres amics

amicable in Danish: Venskabstal

amicable in German: Befreundete Zahl

amicable in Esperanto: Amikebla nombro

amicable in Spanish: Números amigos

amicable in French: Nombre amical

amicable in Galician: Números amigos

amicable in Korean: 친화수

amicable in Icelandic: Vingjarnlegar tölur

amicable in Italian: Numeri amicabili

amicable in Hebrew: מספרים ידידים

amicable in Latin: Numeri amicabiles

amicable in Lithuanian: Draugiškasis
skaičius

amicable in Lombard: Nümar amiis

amicable in Hungarian: Barátságos számok

amicable in Dutch: Bevriende getallen

amicable in Japanese: 友愛数

amicable in Norwegian: Vennskapstall

amicable in Polish: Liczby zaprzyjaźnione

amicable in Portuguese: Número amigável

amicable in Russian: Дружественные числа

amicable in Slovenian: Prijateljsko
število

amicable in Swedish: Vänskapliga tal

amicable in Thai: จำนวนเชิงมิตร

amicable in Chinese: 相亲数

# Synonyms, Antonyms and Related Words

accordant, affable, agreeable, agreeing, akin, amiable, at one, attuned, beneficent, benevolent, benign, benignant, blissful, brotherly, cheerful, civil, compatible, complaisant, concordant, congenial, cooperative, cordial, corresponding, courteous, desirable, dulcet, empathetic, empathic, en rapport, enjoyable, fair, fair and pleasant, favorable, felicific, felicitous, fine, fraternal, frictionless, friendlike, friendly, genial, good, goodly, gracious, grateful, gratifying, harmonious, heart-warming,
honeyed, in accord, in
concert, in rapport, in tune, inharmony, kind, kindly, kindly-disposed, likable, like-minded, mellifluous, mellow, neighborlike, neighborly, nice, of one mind, pacific, peaceable, peaceful, pleasant, pleasing, pleasurable,
pleasure-giving, pleasureful, polite, propitious, rewarding, satisfying, simpatico, sisterly, sociable, sweet, sympathetic, sympathique, together, understanding, unhostile, united, warm, welcome, well-affected,
well-disposed, well-intentioned, well-meaning,
well-meant