Abundancy Ratio Calculator
Determine the abundancy index of any natural number and explore its mathematical properties
Advanced Options
Abundancy Results
Enter a number to determine its classification based on abundancy ratio.
Divisors List
Divisors will appear here when calculated.
Prime Factorization
Prime factors will appear here when calculated.
Comparison with Nearby Numbers
Nearby numbers comparison will appear here.
📚 Mathematical Background
What is Abundancy?
The abundancy index of a positive integer n is the ratio σ(n)/n, where σ(n) is the sum of all divisors of n.
Number Classification
Perfect: σ(n)/n = 2
Abundant: σ(n)/n > 2
Deficient: σ(n)/n < 2
Perfect Numbers
Numbers where σ(n) = 2n (equal to the sum of their proper divisors). The first few are 6, 28, 496, 8128.
Multiplicative Property
The abundancy index is multiplicative: if gcd(a,b)=1, then σ(ab)/ab = σ(a)/a × σ(b)/b.
Calculation Formula
For n = p₁ᵃ × p₂ᵇ × ... × pₖᶻ, σ(n)/n = Π (1 + 1/p + 1/p² + ... + 1/pᵃ) for each prime factor p.
Applications
Used in number theory, studying friendly numbers, perfect numbers, and the distribution of divisors.
🌟 Famous Numbers and Their Abundancy
| Number | Name | Abundancy | Classification |
|---|---|---|---|
| 6 | First Perfect Number | 2 | Perfect |
| 12 | First Abundant Number | 2.333... | Abundant |
| 28 | Second Perfect Number | 2 | Perfect |
| 496 | Third Perfect Number | 2 | Perfect |
| 945 | First Odd Abundant | 2.017... | Abundant |
| 8128 | Fourth Perfect Number | 2 | Perfect |
| 33550336 | Fifth Perfect Number | 2 | Perfect |
Note: This calculator provides exact mathematical results for numbers up to 1,000,000. For larger numbers, factorization may become computationally intensive.