Classification,+Comparison+and+contrast+-+Version+2



 **Great Job... ** **Amicable numbers **  The amicable numbers can be perfect numbers or sociable numbers.  Malba Tahan say **S ** 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. **Super **  ***Sociable 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__s__ divisors of the first one, the third one is the sum of the proper positive__s__ 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.   **Omit the underlined "Ss" **
 * Classification **<span style="font-family: 'Arial','sans-serif'; font-size: 8pt;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt;">*Perfect numbers: **<span style="font-family: 'Arial','sans-serif'; font-size: 8pt;">

<span style="font-family: 'Arial','sans-serif'; font-size: 8pt;"> A number is aspiring if the sum of the proper positive__s__ divisors of the sum of the proper positive__s__ divisors of the sum of the proper positive__s__ divisors, …, etc, of the sum of the proper positive__s__ divisors of this numbers is an amicable number; for example 25, 1 +2 +3 = 6 = 1+5, where 1, 2, and 3 are proper positive__s__ 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__. <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">Confusing... explain why??? ** <span style="font-family: 'Arial','sans-serif'; font-size: 8pt;"> 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) <span style="font-family: 'Arial','sans-serif'; font-size: 8pt;">2 <span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">, where 220 is an amicable odious number. <span style="font-family: 'Arial','sans-serif'; font-size: 8pt;"> A number is called abundant number if the sum of its proper positive divisors is __mayor__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">higher **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">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="font-family: 'Arial','sans-serif'; font-size: 8pt;"> 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="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">also **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">proper positives divisors of 100 <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">( **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">__too__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">informal, omit) **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">. These numbers are the __more__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">"most" because it is a superlative **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">__seem__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">similar **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">to perfect numbers __for__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">by **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">definition and are a class of abundant number. <span style="font-family: 'Arial','sans-serif'; font-size: 8pt; line-height: 115%;"> <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">AN **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">Other class of abundant number is __the__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">omit **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;"> rare numbers. T__he__ 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. <span style="color: #92d050; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: 115%;">=-)! <span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;"> These are a __numbers more different__ <span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">(order) more different numbers **<span style="color: black; font-family: 'Arial','sans-serif'; font-size: 12pt;"> **<span style="color: #ff0000; font-family: 'Arial','sans-serif'; font-size: 12pt;">than **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">__to__ perfect numbers for definition, but, these are defined with the same terms than it. <span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">* <span style="color: blue; font-family: 'Arial','sans-serif'; font-size: 12pt;">[] <span style="color: red; font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * <span style="color: #b2a1c7; font-family: 'Arial','sans-serif'; font-size: 12pt;">Comparison and Contrast **
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt;">*Aspiring number: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt;">*Untouchable numbers: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt;">*Abundant numbers: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt;">*Partially perfect numbers: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * <span style="color: #92cddc; font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">*Rare number: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt; line-height: 115%;">
 * <span style="color: #b2a1c7; font-family: 'Arial','sans-serif'; font-size: 12pt;">Bibliography: **<span style="font-family: 'Arial','sans-serif'; font-size: 12pt;">
 * “Matemática divertida y curiosa ” by Malba Tahan. 2007, Intermedio Ediciones Ltda. Bogotá, Colombia. Pp. 23, 24 and 65.
 * <span style="color: blue; font-family: 'Arial','sans-serif'; font-size: 12pt;">[] <span style="color: red; font-family: 'Arial','sans-serif'; font-size: 8pt;">

Guyyyyyyyyy!!!!!!! =-)!!!