Classification,+Comparison+and+contrast+-+Version+1+with+classmate+comments



** Amicable numbers ** ** Classification ** The amicable numbers can be perfect numbers or sociable numbers. There was whom inferred that if a number was the sum of its own proper positives divisor (positive divisors should be written), it was “egoistic numbers” (it's better written if you say an "egoistic number" because does not fit with the previous lines where you talk about "a number") has say Malba Tahan in his book (It's not very clear what exactly Malba Than said on the book if all the previous lines or just the naming of the egoistic numbers). But It not exist egoistic numbers (I will be honest, i don't truly understand what are you trying to say here, no offense, of course. But if you wanna say that the egoistic numbers does not exist better try writting "But there are not egoistic numbers"). If a amic able (just common mistake, write amicable together) 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. 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 positives divisors of the first one, the third one is the sum of the proper positives 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.
 * *Perfect numbers: **
 * Sociable numbers:**

** Comparison and Contrast ** A number is aspiring if the sum of the proper positives divisors of the sum of the proper positives divisors of the sum of the proper positives divisors, …, etc, of the sum of the proper positives divisors of this numbers is an amicable number; for example 25, 1 +2 +3 = 6 = 1+5, where 1, 2, and 3 are proper positives divisors of 6, and 1 and 5 are proper positives divisors of 25. However these numbers can be called perfect amicable numbers but they are not it properly. (Very well done, i like it) The numbers that are not amicable are not odious numbers, they are called untouchable numbers (A funny way to contrast! Congratulations!). The odious numbers has not similarity whit amicable numbers (wait, you said that the numbers that are not amicalbe, and aren't either odious, why are again called odious? i don't get it. They shouldn't be named untouchable?). The odious numbers are numbers whose binary base expression has an unpaired amount of ones; for example 21= 101012. On the contrary of amicable numbers, the untouchable numbers are the numbers that are not the sum of the proper positive divisors of any other one; for example, 2 and 5 are untouchable numbers. (I'm truly confused on all this lines, i don't have clear what are the odious and what are the untouchable numbers, please ty to explain this better) A numbers (Number, not numbers, does not fit with the next words that talks in single, not plural) is called abundant number if the sum of its proper positive divisors is mayor than it. Then a (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 it (the previous parts are called in plural, and this one is not) can not be perfect 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 proper positives divisors of 100 too. These numbers are the more seem to perfect numbers for definition and are a class of abundant number. Other class of abundant number is the rare numbers. The 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. =-)! This is a definition more (I think very fits beter) different to perfect numbers that use the same terms of that (it) . ** Bibliography: ** 
 * *Aspiring number: **
 * *Untouchable numbers: **
 * *Abundant numbers: 2.3.5.2 **
 * *Partially perfect numbers: **
 * *Rare number: **
 * “Matemática divertida y curiosa ” by Malba Tahan. 2007, Intermedio Ediciones Ltda. Bogotá Colombia. Pp. 23, 24 and 65.
 * []
 * [] 


 * Super.... Let's see what Juan thinks... Please correct Juan's paper... **