Perfect Number Calculator
Perfect, abundant, or deficient with divisors.
What it does
Perfect numbers balance exactly: divisors sum back to the number, as 1+2+4+7+14 rebuilds 28. Abundant overshoot while deficient fall short, and only four small perfects are known to most.
The tool above takes n to 100000. It returns the verdict with divisors and the next known perfect.
The default 28 reads perfect against 496 next. The 6 case reads perfect toward 28, 12 reads abundant, and 8 reads deficient.
Aliquot sum definitions classify every integer with Euclid's even perfect form behind them. Prime hunters chase the Mersenne primes that generate each even perfect.
Odd perfects may not exist at all: none found through vast searches. Primes always read deficient since only 1 divides them properly.
How to use
- Type n from 1 to 100000.
- Read the verdict first.
- Read divisors beside it.
- Read next perfect last.
- Compare abundant cousins.
- Respect the 100000 cap.
Formula
- n
- Integer 1 to 100000.
- s(n)
- Proper divisor sum.
Number theory defines perfect numbers through aliquot sums with Euclid's even form tied to Mersenne primes. GIMPS hunts the underlying Mersenne primes behind each even perfect. OEIS A000396 tabulates 6, 28, 496, 8128 for the next lookup.
Worked example
Example: is 28 perfect
Default: 28.
- Number: 28.
- Divisors: 1, 2, 4, 7, 14 (proper).
- Sum: 1 + 2 + 4 + 7 + 14 = 28.
- Verdict: sum equals number, so perfect.
- Next: 496 follows 28.
- Tool shows Perfect, 1 + 2 + 4 + 7 + 14 = 28, Next perfect: 496.
Reference table
| n | Divisors | Sum |
|---|---|---|
| 6 | 1 + 2 + 3 | 6 |
| 28 | 1 + 2 + 4 + 7 + 14 | 28 |
| 496 | 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 | 496 |
| 8128 | long list | 8128 |
Limits
- n to 100000 for speed.
- Even perfects listed four.
- Odd unknown flagged past.
- Divisor lists grow long.
Common mistakes
- Counting n itself as proper.
- Reading abundant as broken.
- Expecting many perfects.
- Skipping the next lookup.
- Trusting odd perfect claims.
Try it yourself
Quick checks. Answers sit tucked under each question.
6?
Perfect with 1 + 2 + 3 = 6, next 28.
12?
Abundant with 1 + 2 + 3 + 4 + 6 = 16.
8?
Deficient with 1 + 2 + 4 = 7.
0?
Zero divides nothing: enter 1 to 100000.
Questions
Perfect meaning?
Divisors sum to the number: 6 = 1+2+3, 28 = 1+2+4+7+14. Euclid links them to Mersenne primes.
Other verdicts?
Over sums abundant, under sums deficient. 12 runs abundant at 16, 8 deficient at 7.
Known perfects?
6, 28, 496, 8128 known. Odd perfects remain unproven either way.
Factor how?
Yes, through square root pairing. This tool factors to 100000 fast.
Huge lists shown?
No, divisor lists grow long past thousands. Verdicts stay exact.
All numbers classify?
Yes, aliquot sums classify every integer. Primes always read deficient.