Everything about Amicable Number totally explained
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 period2.
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 can't 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 isn't known whether an even-odd pair of amicable numbers exists. Also, every known pair shares at least one common factor. It isn't 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 can't 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's 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):
sum_x=0
sum_y=0
for i in range(1,x):
if x%i==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
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
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