Difference between revisions of "Manuals/calci/PERFECTNUMBERS"
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*Perfect number is a positive integer which is equal to the sum of its proper divisors. | *Perfect number is a positive integer which is equal to the sum of its proper divisors. | ||
*The smallest perfect number is 6, which is the sum of 1, 2, and 3. | *The smallest perfect number is 6, which is the sum of 1, 2, and 3. | ||
| + | |||
| + | ==ABUNDANTNUMBERS== | ||
| + | *Abundant number is a number for which the sum of its proper divisors is greater than the number itself. | ||
| + | *It is also called Excessive Number. | ||
| + | *The first Abundant number is 12. | ||
| + | * Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. | ||
| + | *The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4. | ||
| + | *ABUNDANTNUMBERS(110) = 12 18 20 24 30 36 40 42 48 54 56 60 66 70 72 78 80 84 88 90 96 100 102 104 108 | ||
| + | |||
| + | ==DEFICIENTNUMBERS== | ||
| + | *Deficient number is a number <math>n</math> for which the sum of divisors <math>\sigma(n)<2n</math>, or, equivalently, the sum of proper divisors (or aliquot sum) <math>s(n)<n</math>. The value <math>2n</math> − <math>\sigma(n)</math> (or <math>n</math> − <math>s(n)</math>) is called the number's deficiency. | ||
| + | *Properties of Deficient Numbers are: | ||
| + | *The aliquot sums of prime numbers equal 1, all prime numbers are deficient. | ||
| + | *An infinite number of both even and odd deficient numbers exist. | ||
| + | *All odd numbers with one or two distinct prime factors are deficient. | ||
| + | *All proper divisors of deficient or perfect numbers are deficient. | ||
| + | *Some of the Deficient numbers are: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17 and etc. | ||
| + | *DEFICIENTNUMBERS(110) = 2 3 4 5 7 8 9 10 11 13 14 15 16 17 19 21 22 23 25.......... | ||
==Examples== | ==Examples== | ||
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#PERFECTNUMBERS(900,54) = 6 28 496 | #PERFECTNUMBERS(900,54) = 6 28 496 | ||
#PERFECTNUMBERS(-567,0) = NaN | #PERFECTNUMBERS(-567,0) = NaN | ||
| + | #PERFECTNUMBERS(10) = 6 | ||
| + | |||
| + | ==Related Videos== | ||
| + | |||
| + | {{#ev:youtube|v=8QdCLM8Zqas|280|center|Perfect Numbers}} | ||
==See Also== | ==See Also== | ||
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==References== | ==References== | ||
[[https://www.britannica.com/topic/perfect-number Perfect number]] | [[https://www.britannica.com/topic/perfect-number Perfect number]] | ||
| + | |||
| + | |||
| + | |||
| + | *[[Z_API_Functions | List of Main Z Functions]] | ||
| + | |||
| + | *[[ Z3 | Z3 home ]] | ||
Latest revision as of 03:14, 11 February 2020
PERFECTNUMBERS (Upto,StartFrom)
- is the maximum number limit.
- is the minimum number limit.
is the maximum number limit.
is the minimum number limit.Description
- This function returns the list of Perfect numbers.
- In PERFECTNUMBERS (Upto,StartFrom),Upto is the Maximum number limit to display and Start from is the Minimum number to start Perfect number list.
- Perfect number is a positive integer which is equal to the sum of its proper divisors.
- The smallest perfect number is 6, which is the sum of 1, 2, and 3.
ABUNDANTNUMBERS
- Abundant number is a number for which the sum of its proper divisors is greater than the number itself.
- It is also called Excessive Number.
- The first Abundant number is 12.
- Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16.
- The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4.
- ABUNDANTNUMBERS(110) = 12 18 20 24 30 36 40 42 48 54 56 60 66 70 72 78 80 84 88 90 96 100 102 104 108
DEFICIENTNUMBERS
- Deficient number is a number for which the sum of divisors , or, equivalently, the sum of proper divisors (or aliquot sum) . The value − (or − ) is called the number's deficiency.
- Properties of Deficient Numbers are:
- The aliquot sums of prime numbers equal 1, all prime numbers are deficient.
- An infinite number of both even and odd deficient numbers exist.
- All odd numbers with one or two distinct prime factors are deficient.
- All proper divisors of deficient or perfect numbers are deficient.
- Some of the Deficient numbers are: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17 and etc.
- DEFICIENTNUMBERS(110) = 2 3 4 5 7 8 9 10 11 13 14 15 16 17 19 21 22 23 25..........
for which the sum of divisors 
, or, equivalently, the sum of proper divisors (or aliquot sum) 
. The value 
− 
(or 
) is called the number's deficiency.Examples
- PERFECTNUMBERS(14,2) = 6
- PERFECTNUMBERS(900,54) = 6 28 496
- PERFECTNUMBERS(-567,0) = NaN
- PERFECTNUMBERS(10) = 6
Related Videos
See Also
References