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Numeros hemimperfectos

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Some numbers , like perfect , semiperfect , abundant and deficient , defined from the positive or rather from the sum of the specific factors that make each group dividers. Thus we have a number is perfect as long as the sum of all its proper divisors is equal to the given number ; say that a number is deficient if the sum of all its proper divisors is less than the number given , if such sum is greater than the given number say that the number is abundant. The dividers also define the semiperfect numbers, which are defined as those that can be expressed as the sum of some of its own splitters. Considering the properties that can be found among the divisors of a given number, in this written hemimperfects numbers (prefix hemi - Greek meaning half) is. Vaguely defined, we can say that a number is hemimperfect if it can be expressed as the sum of half of its divisors.

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Numeros hemimperfectos

  1. 1. Números Hemimperfectos José Acevedo Jiménez Santiago, Rep. Dom. divulgadoresrd@hotmail.com Abstract. Algunos números, como los perfectos, semiperfectos, abundantes y deficientes, se definen a partir de sus divisores positivos o más bien a partir de la suma de los factores propios que conforman a cada grupo. Así tenemos que un número es perfecto siempre y cuando la suma de todos sus divisores propios sea igual al número dado; decimos que un números es deficiente si la suma de todos sus divisores propios es menor al número dado, si tal suma es mayor que el número dado decimos que el número es abundante. Los divisores también definen a los números semiperfectos, que se definen como aquellos que se pueden expresar como la suma de algunos de sus divisores propios. Atendiendo las propiedades que podemos encontrar entre los divisores de un número dado, en el presente escrito se muestran los números hemimperfectos (hemi- prefijo griego que significa mitad). Definidos de una manera vaga, podemos decir que un número es hemimperfecto si se puede expresar mediante la suma de la mitad de sus divisores, es decir, si un número posee una cantidad de divisores (incluidos la unidad y el propio número) y tomando solo , a partir de un algoritmo conocido, es posible sumar el número dado, entonces, podemos tener la certeza de que tal número es hemimperfecto. Some numbers , like perfect , semiperfect , abundant and deficient , defined from the positive or rather from the sum of the specific factors that make each group dividers. Thus we have a number is perfect as long as the sum of all its proper divisors is equal to the given number ; say that a number is deficient if the sum of all its proper divisors is less than the number given , if such sum is greater than the given number say that the number is abundant. The dividers also define the semiperfect numbers, which are defined as those that can be expressed as the sum of some of its own splitters. Considering the properties that can be found among the divisors of a given number, in this written hemimperfects numbers (prefix hemi - Greek meaning half) is. Vaguely defined, we can say that a number is hemimperfect if it can be expressed as the sum of half of its divisors. Keywords. Divisores propios, divisores, números hemimperfectos, números perfectos, números abundantes, números defectivos, números semiperfectos, número de Fermat. Un número es hemimperfecto si tiene la forma: Entre sus divisores se encuentran los números: , y . Donde:
  2. 2. En la tabla subsiguiente se muestran algunos números hemimperfectos, para valores de . 0 1 2 3 4 3 5 17 257 65537 12 40 544 131584 8590065664 Diagrama de flujo para saber si un número es hemimperfecto. Nota: el número impar debe ser de la forma: . Es decir un número de Fermat.
  3. 3. Algunas características de los números hemimperfectos. Los números hemimperfectos son números semiperfectos. Eso significa que tales números se pueden expresar como la suma de algunos de sus divisores propios. Ejemplo: Los divisores propios del 40 son: 20, 10, 8, 5, 4, 2, 1. Como se puede observar, entre los divisores propios del 40 se encuentran, por lo menos, dos números pares consecutivos (10, 8). Esta característica es distintiva, pero no exclusiva, de los números hemimperfectos. Los divisores propios que sumados nos dan el número hemimperfecto son fáciles de encontrar. Ejemplo: Dado el número hemimperfecto 544, encontrar los divisores que sumados sean iguales al número dado. Para encontrar tales divisores, hacemos lo siguiente: Dividimos entre dos el número dado. El resultado también lo dividimos entre dos y repetimos el proceso hasta que el resultado sea un número impar (primo). Al último resultado le restamos dos, este número también es un divisor propio del número dado; finalmente sumamos todos los divisores encontrados y a dicha suma le agregamos dos.
  4. 4. Los divisores propios encontrados de 544 son: La suma de los divisores propios encontrados es: . Si , entonces: Donde: Todo número hemimperfecto tiene una cantidad par de divisores (incluidos el uno y el propio número). Del número total de divisores positivos , sólo basta tomar (ni más ni menos) para que, al ser sumados, nos den el número hemimperfecto dado. De ahí el nombre hemimperfecto (hemi- prefijo griego que significa mitad). Si , entonces: Divisores no exclusivos. Los divisores , y no son exclusivos de los números hemimperfectos. Por ejemplo, el número 72 posee entre sus divisores los
  5. 5. números: , y . Pese a tener los mencionados divisores, el 72 no es un número hemimperfecto. Esto se puede verificar al buscar los divisores . ; 16 no es un divisor de 72, por lo tanto, 72 no es un número hemimperfecto. Otro ejemplo es el 144, al buscar los divisores , tenemos que: 144/2 = 72 72/2 = 36 36/2 = 18 18/2 = 9 18 – 2 = 16 En este caso vemos que el 16, efectivamente, divide al 144. Eso nos puede hacer pensar que el 144 es un número hemimperfecto, pero, no lo es. Dado que el 9, número impar, no puede ser expresado como: . Como se ha podido observar los números hemimperfectos poseen propiedades distintivas que los distinguen de otros conjuntos numéricos. Si bien no todas son exclusivas de ellos, la combinación de sus propiedades lo hace un conjunto distinguible y clasificable.
  6. 6. Referencias. Carmichael, R. (1907). A table of multiply perfect numbers. Bull. Amer. Navarro Loidi, Juan. (2002). Los “Elementos” de Euclides. Jara, Pascual y Villegas, salvador. (2008). Números Curiosos. www.ugr.es/~anillos/textos/pdf/2008/numeros.pdf Mora Flores, Walter. (2012). Introducción a la Teoría de Números. Ejemplos y algoritmos. 1ra ed. (versión digital).

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