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**Great Job** Amicable numbers are two integer numbers . IF i f one is the sum of the proper positive divisors of the other one, the divisor major than zero and minor than the number, and vice versa, this is, if //‘a’ // and //‘b’ // are amicable numbers; //‘b’ // is the sum of the proper positives divisors of //‘a’ // and //‘a’ // is the sum of the proper positive divisors of //‘b’ //. For example: If we search the positive divisors of 220 we get:  1x220 = 2x110 = 4x55 = 5x44 = 10x22 = 11x20 If we extract the proper divisors and we add that: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 If we search the positives divisors of 284 we get: 1 x 284 = 2x142 = 4x71 If we extract the proper divisors and we add that: 1 + 2 + 4 + 71 + 142 = 220 Then 220 and 248 are a pair amicable numbers, knew by the Pythagoreans.
 * Amicable numbers **
 * Definition **
 * 220 **
 * 284 **

The Pythagoreans thought that these amicable numbers, 220 and 284 ha__s__**had** mystical properties. The people thought that they should make two persons become friends using those properties, giving them food **whit** characteristics of two amicable numbers in different place**S** at the same time. Some amicable numbers are easy to find with the formula discovered by Leonhard Euler (1707-1783) in the year 1750, which is absurdly attributed to Thabit ibn Qurra (826-901), sometimes to René Descartes (1596-1650), or Pierre de Fermat (1601-1665), whom discovered three pairs of amicable numbers ( 6232, 6368 ; 9.363.584, 9.437.056 (1638); 17.296, 18.416  (1636)) correspondently. The formula is: If //p = //3 × 2//n //-1 1, //<span style="font-family: 'Arial','sans-serif';">q = //3 × 2//<span style="font-family: 'Arial','sans-serif';">n //  1, //<span style="font-family: 'Arial','sans-serif';">r // = 9 × 22//<span style="font-family: 'Arial','sans-serif';">n //-1  1, where //<span style="font-family: 'Arial','sans-serif';">n // > 1 is an integer and //<span style="font-family: 'Arial','sans-serif';">p //, //<span style="font-family: 'Arial','sans-serif';">q //, and //<span style="font-family: 'Arial','sans-serif';">r // are prime numbers, then 2//<span style="font-family: 'Arial','sans-serif';">npq // and 2//<span style="font-family: 'Arial','sans-serif';">nr // are a pair of amicable numbers. Euler discovered 60 pairs of amicable numbers with his formula in 1750. But the second pair ( 1184, 1210 ) was ignored by the scientists and was discovered in 1866 by Niccolò Paganini, an Italian boy with 16 years old. The pair 6232, 6368 can not be found with the formula.
 * <span style="color: #b2a1c7; font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">Description **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">

<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;"> The amicable numbers can be perfect numbers or sociable numbers.
 * <span style="color: #b2a1c7; font-family: 'Arial','sans-serif';">Classification **

Malba Tahan says in his book than there was whom inferred than if a number was the sum of its own proper positive divisors, it was “egoistic number”. But there are not egoistic numbers. If an amicable number is the sum of its own proper positives divisor, it is called a perfect number; for example 6, 6= 1 x 6= 2 x3 where 1 + 2 + 3= 6.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Perfect numbers: **

If a amicable number is amicable in joint with more than only one number, it is a sociable number, then the second number is the sum of the proper positive divisors of the first one, the third one is the sum of the proper positive divisors of the second one, and so on the first one is the sum of the proper positive divisors of the last one; for example 12496, 14288, 15472, 14536 y 14264.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Sociable numbers: **


 * <span style="color: #b2a1c7; font-family: 'Arial','sans-serif';">Comparison and Contrast **

A number is aspiring if the sum of the proper positive divisors of the sum of the proper positive divisors of the sum of the proper positive divisors, …, etc, of the sum of the proper positive divisors of this numbers is a perfect number; for example 25, 1 +2 +3 = 6 = 1+5, where 1, 2, and 3 are proper positive divisors of 6, and 1 and 5 are proper positive__s__ divisors of 25. However these numbers can be called perfect amicable numbers but they are not it properly because the sum of the proper positive divisors of an aspiring number is not the same number.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Aspiring number: **

The numbers that are not amicable are not odious numbers, they are called untouchable numbers. The untouchable numbers are the numbers that are not the sum of the proper positive divisors of any other one, then these are not amicable numbers; for example, 2 and 5 are untouchable numbers while, the odious numbers can be amicable numbers because odious numbers are numbers whose binary base expression has an unpaired amount of ones; for example 220, 220= (11011100)2, where 220 is an amicable odious number.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Untouchable numbers: **

A number is called abundant number if the sum of its proper positive divisors is <span style="font-family: 'Arial','sans-serif';">higher than it. Then an abundant number is not a perfect number; for example 220, 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284> 220. Then the abundant numbers can be amicable numbers, however these can not be perfect numbers.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Abundant numbers: **

If a number is the sum of some of its proper positive divisors, them it is a partially perfect number; for example 100, 10 + 20 + 25 + 50 = 105>100 but 1, 2, 4, 5, etc, are <span style="font-family: 'Arial','sans-serif';">also proper positives divisors of 100. These numbers are the <span style="font-family: 'Arial','sans-serif';">most similar to perfect numbers <span style="font-family: 'Arial','sans-serif';">by  definition and are a class of abundant number.
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Partially perfect numbers: **

<span style="font-family: 'Arial','sans-serif';">An other class of abundant number is rare numbers. Rare numbers are the abundant numbers that are not the sum of some of its proper positives divisors. They are the abundant numbers that are not partially perfect numbers; for example 70 and 836 are rare numbers. =-)! These are a more different numbers <span style="font-family: 'Arial','sans-serif';">than perfect numbers for definition, but, these are defined with the same terms than it. <span style="color: #b2a1c7; font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">**Bibliography:** <span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif';">*Rare number: **
 * “Matemática divertida y curiosa ” by Malba Tahan. 2007, Intermedio Ediciones Ltda. Bogotá, Colombia. Pp. 23, 24 and 65.
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<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">=-)!!!