Manuals/calci/PERFECTNUMBERS

PERFECTNUMBERS (Upto,StartFrom)

• $Upto$ is the maximum number limit.
• $StartFrom$ 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 $n$ for which the sum of divisors $\sigma (n)<2n$, or, equivalently, the sum of proper divisors (or aliquot sum) $s(n)<n$. The value Failed to parse (syntax error): {\displaystyle 2n − \sigma(n)} (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n − s(n)} ) is called the number's deficiency.

Examples

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

Related Videos

Perfect Numbers

See Also

• SUM
• AVERAGE
• AVERAGEA

References

[Perfect number]

• List of Main Z Functions

• Z3 home