Calculator Form
Use single mode for detailed divisor analysis. Use range mode to compare many values and visualize aliquot sums across an interval.
Example Data Table
These sample values illustrate common aliquot sum patterns.
| Number | Proper Divisors | Aliquot Sum | Classification |
|---|---|---|---|
| 1 | None | 0 | Deficient |
| 6 | 1, 2, 3 | 6 | Perfect |
| 8 | 1, 2, 4 | 7 | Deficient |
| 12 | 1, 2, 3, 4, 6 | 16 | Abundant |
| 20 | 1, 2, 4, 5, 10 | 22 | Abundant |
| 28 | 1, 2, 4, 7, 14 | 28 | Perfect |
Formula Used
The aliquot sum of a positive integer n is the sum of all its proper divisors. Proper divisors are positive divisors smaller than the number itself.
Aliquot Sum: s(n) = Σ d, for every proper divisor d of n.
Equivalent Form: s(n) = σ(n) − n, where σ(n) is the sum of all positive divisors of n.
Classification Rules:
- Deficient: s(n) < n
- Perfect: s(n) = n
- Abundant: s(n) > n
Example for 12: proper divisors are 1, 2, 3, 4, and 6. Therefore, s(12) = 1 + 2 + 3 + 4 + 6 = 16.
How to Use This Calculator
- Select Single Number for one detailed result, or Range Scan for many values.
- Enter the integer you want to evaluate, or set the start, end, and step for a range.
- Choose the aliquot sequence length to trace repeated sums.
- Optionally filter range results by deficient, perfect, or abundant numbers.
- Click Calculate Aliquot Sum to show the results above the form.
- Review the summary table, divisors, classification, sequence, and Plotly graph.
- Use the CSV and PDF buttons to export result data or example tables.
FAQs
1. What is an aliquot sum?
An aliquot sum is the sum of all proper divisors of a positive integer. Proper divisors divide the number exactly and exclude the number itself.
2. What are proper divisors?
Proper divisors are positive integers smaller than the original number that divide it without leaving a remainder. For 10, they are 1, 2, and 5.
3. Why does 1 have an aliquot sum of 0?
The number 1 has no positive proper divisors because the only divisor of 1 is itself. Since proper divisors exclude the number, the sum is zero.
4. What is a perfect number?
A perfect number has an aliquot sum equal to the number itself. For example, 6 is perfect because 1 + 2 + 3 = 6.
5. What is an abundant number?
An abundant number has an aliquot sum greater than the original number. For instance, 12 is abundant because its proper divisors add up to 16.
6. What is a deficient number?
A deficient number has an aliquot sum smaller than the original number. Prime numbers are deficient because their only proper divisor is 1.
7. What is an aliquot sequence?
An aliquot sequence starts with a number and repeatedly replaces it with its aliquot sum. Sequences may end at zero, repeat, or cycle.
8. Why use range mode?
Range mode helps compare many integers quickly. It is useful for spotting perfect, abundant, or deficient numbers and for visualizing how aliquot sums change across intervals.