1 disposed to please; "an amiable villain with a cocky sidelong grin"- Hal Hinson [syn: good-humored, good-humoured]
2 diffusing warmth and friendliness; "an affable smile"; "an amiable gathering"; "cordial relations"; "a cordial greeting"; "a genial host" [syn: affable, cordial, genial]
EtymologyFrom French amiable, from Latin amicabilis ("friendly"), from amicus ("friend"), from amare ("to love"). The meaning has been influenced by French aimable, Latin amabilis ("lovable"), from amare ("to love"). Compare with amicable, amorous, amability, Latin amat love.
- a US /ˈeɪ.mi.ə.bəl/
possessing sweetness of disposition
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,
- q = 3 × 2n − 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%k0: sum_y+=k return sum_x
y and sum_yx
- 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
- Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (pp. 145 - 147). London: Penguin Group.
amiable in Belarusian (Tarashkevitsa): Прыхільныя лікі
amiable in Breton: Niver karantezus
amiable in Catalan: Nombres amics
amiable in Danish: Venskabstal
amiable in German: Befreundete Zahl
amiable in Esperanto: Amikebla nombro
amiable in Spanish: Números amigos
amiable in French: Nombre amical
amiable in Galician: Números amigos
amiable in Korean: 친화수
amiable in Icelandic: Vingjarnlegar tölur
amiable in Italian: Numeri amicabili
amiable in Hebrew: מספרים ידידים
amiable in Latin: Numeri amicabiles
amiable in Lithuanian: Draugiškasis skaičius
amiable in Lombard: Nümar amiis
amiable in Hungarian: Barátságos számok
amiable in Dutch: Bevriende getallen
amiable in Japanese: 友愛数
amiable in Norwegian: Vennskapstall
amiable in Polish: Liczby zaprzyjaźnione
amiable in Portuguese: Número amigável
amiable in Russian: Дружественные числа
amiable in Slovenian: Prijateljsko število
amiable in Swedish: Vänskapliga tal
amiable in Thai: จำนวนเชิงมิตร
amiable in Chinese: 相亲数
accommodating, affable, affectionate, agreeable, amicable, approachable, benevolent, benign, benignant, blissful, bonhomous, brotherly, cheerful, civil, clubbable, clubbish, clubby, communicative, companionable, companionate, compatible, complaisant, compliant, congenial, cordial, courteous, decent, desirable, dulcet, easy, easy-natured, en rapport, enjoyable, fair, fair and pleasant, favorable, felicific, felicitous, fine, fit for society, fond of society, fraternal, friendlike, friendly, generous, genial, gentle, good, good-hearted, good-humored, good-natured, good-tempered, goodly, gracious, grateful, gratifying, gregarious, harmonious, heart-warming, hearty, honeyed, hospitable, indulgent, kind, kindly, lenient, liberal, likable, mannerly, mellifluous, mellow, mild, neighborlike, neighborly, nice, obliging, open, openhearted, overindulgent, overpermissive, peaceable, permissive, pleasant, pleasing, pleasurable, pleasure-giving, pleasureful, receptive, responsive, rewarding, satisfying, simpatico, sisterly, sociable, social, social-minded, sweet, sweet-tempered, sympathique, tractable, unhostile, urbane, warm, warmhearted, welcome, welcoming, well-affected, well-disposed, well-intentioned, well-meaning, well-meant, well-natured, winning, winsome