Compute divisor totals for any positive integer instantly. Inspect factors, counts, and proper sums easily here. Visualize divisor growth and export polished reports with confidence.
Calculate the sum of all divisors of a positive integer, inspect prime factorization, review proper divisors, export clean reports, and visualize divisor behavior with an interactive chart.
| Input n | Prime Factorization | σ(n) | Proper Divisor Sum | Divisor Count | Classification |
|---|---|---|---|---|---|
| 8 | 23 | 15 | 7 | 4 | Deficient |
| 12 | 22 × 3 | 28 | 16 | 6 | Abundant |
| 28 | 22 × 7 | 56 | 28 | 6 | Perfect |
| 36 | 22 × 32 | 91 | 55 | 9 | Abundant |
If a number is written as a prime factorization n = p1a1 × p2a2 × ... × pkak, then the sum of divisors function is:
This equals the product of the geometric-series sums for each prime factor. The calculator also finds:
It means the total of every positive divisor of a number, including 1 and the number itself. For 12, the divisors are 1, 2, 3, 4, 6, and 12, so the sum is 28.
A proper divisor sum excludes the number itself. For 12, the proper divisors are 1, 2, 3, 4, and 6. Their sum is 16, which is smaller than the full divisor sum of 28.
Prime factorization makes divisor calculations faster. Instead of listing every divisor manually, the calculator uses the factorization formula to compute the sum of divisors and divisor count efficiently.
A perfect number equals the sum of its proper divisors. For example, 28 has proper divisors 1, 2, 4, 7, and 14, and their sum is 28.
An abundant number has a proper divisor sum greater than the number itself. For example, 12 is abundant because its proper divisor sum is 16.
A deficient number has a proper divisor sum smaller than the number. Prime numbers are always deficient because their only proper divisor is 1.
Some numbers have many divisors. Generating every divisor can become heavy for large divisor counts, so the calculator skips full list generation when the total count is very high.
The exports include the main calculation summary, such as the input value, divisor sum, proper divisor sum, divisor count, average divisor, classification, and factorization preview.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.