Calculator Inputs
Use the required number field for a direct checker result. Add an optional range to review nearby abundant, perfect, and deficient values.
Example Data Table
| Number | Proper Divisors | Aliquot Sum | Classification | Abundance or Deficiency |
|---|---|---|---|---|
| 12 | 1, 2, 3, 4, 6 | 16 | Abundant | +4 |
| 18 | 1, 2, 3, 6, 9 | 21 | Abundant | +3 |
| 20 | 1, 2, 4, 5, 10 | 22 | Abundant | +2 |
| 28 | 1, 2, 4, 7, 14 | 28 | Perfect | 0 |
| 15 | 1, 3, 5 | 9 | Deficient | -6 |
Formula Used
- Proper divisors: All positive divisors of n excluding n itself.
- Aliquot sum: s(n) = sum of proper divisors of n.
- Abundant number test: If s(n) > n, the number is abundant.
- Perfect number test: If s(n) = n, the number is perfect.
- Deficient number test: If s(n) < n, the number is deficient.
- Abundance value: s(n) - n.
- Total divisor sum: σ(n) = s(n) + n.
- Abundancy index: σ(n) / n.
These measures help compare divisor-rich integers, inspect factor structure, and identify whether a value has more proper divisor mass than the number itself.
How to Use This Calculator
- Enter the positive integer you want to check.
- Optionally add a start and end range for broader analysis.
- Set the sample result limit for displayed abundant numbers.
- Click Check Number to generate the result block.
- Review classification, divisors, factorization, abundance value, and index.
- Use the export buttons to download the analysis as CSV or PDF.
For teaching or research, range analysis helps compare how abundant values appear across intervals while the single-number summary explains the exact divisor behavior.
Frequently Asked Questions
1. What is an abundant number?
An abundant number has a proper divisor sum greater than the number itself. For example, 12 is abundant because 1 + 2 + 3 + 4 + 6 = 16.
2. What are proper divisors?
Proper divisors are all positive divisors except the number itself. For 20, the proper divisors are 1, 2, 4, 5, and 10.
3. How is abundance value calculated?
The abundance value equals the aliquot sum minus the number. A positive result means abundant, zero means perfect, and a negative result means deficient.
4. What does abundancy index mean?
The abundancy index is σ(n) divided by n. It measures total divisor weight relative to the number and helps compare integers of different sizes.
5. Why does the calculator include perfect and deficient results?
A complete divisor analysis naturally classifies numbers into abundant, perfect, or deficient groups. This makes the checker more useful for comparisons and study.
6. What is a semiperfect status check?
A semiperfect check tests whether some proper divisors can sum exactly to the original number. Many abundant numbers are semiperfect, but weird numbers are not.
7. Why might the semiperfect test be skipped?
Subset-sum testing grows harder for larger values. The calculator skips some large cases to keep performance smooth and prevent slow page responses.
8. Can I export both single and range results?
Yes. When range inputs are filled, the CSV and PDF downloads include the main number analysis plus the optional range summary.