# Manuals/calci/PERFECTNUMBERS

PERFECTNUMBERS (Upto,StartFrom)

• 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..........

## Examples

1. PERFECTNUMBERS(14,2) = 6
2. PERFECTNUMBERS(900,54) = 6 28 496
3. PERFECTNUMBERS(-567,0) = NaN
4. PERFECTNUMBERS(10) = 6

Perfect Numbers