AtlasCalcsUtility Calculation Suite
2026-09-29 8 min

Perfect Number Calculator

Test whether n is perfect, deficient, or abundant with the full aliquot list, σ(n), and s(n).

Calculator

Perfect Number Inputs

Integer (n)
Results update live as you type
Status:s(28) = 28 vs 28PERFECT

Perfect Number Summary

Verdictperfect
Proper divisors1, 2, 4, 7, 14
s(n) vs n28 vs 28

Results

Optimal

Classification

perfect

perfect / deficient / abundant

Aliquot sum s(n)

28

Sum of proper divisors

Sigma σ(n)

56

Sum of all divisors

Proper divisor count

5

How many proper divisors

Gap s(n) − n

0

0 = perfect; − = deficient; + = abundant

Classification by complete divisor-pair enumeration to √n; s(n), σ(n), and every proper divisor listed.

Visualizations

Proper divisors

11
22
44
77
1414

Cumulative aliquot build-up

s(n) relative to 2n

s(n) / 2n28

s(n) vs n

s(n)28

Formula

The aliquot sum adds every proper divisor; its relation to n classifies the number.

s(n) = σ(n) − n; perfect ⟺ s(n) = n

Alternative Form

Even perfects: 2^(p−1)(2^p − 1) with 2^p − 1 Mersenne prime

Variables

SymbolNameUnitDescription
nInputintegerPositive integer 1–10,000,000
s(n)Aliquot sumintegerSum of proper divisors
σ(n)SigmaintegerSum of all divisors = s(n)+n
gaps(n) − ninteger0 perfect; negative deficient; positive abundant

Explanation

Divisor bars stack each proper divisor; the stack height next to n shows the verdict at a glance.

Derivation

Enumerate divisor pairs (d, n/d) to √n, exclude n, sum for s(n), add n for σ(n). Euclid–Euler characterizes even perfects via Mersenne primes.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Read the input.

Input: n
n:28
Result: n = 28
Step 2

List proper divisors by scanning pairs to √n.

d | n, d < n
Intermediate: 1 + 2 + 4 + 7 + 14
Result: 5 proper divisor(s)
Step 3

Add them to get the aliquot sum.

s(n) = Σ proper divisors
Intermediate: 1 + 2 + 4 + 7 + 14
Result: s(28) = 28
Step 4

Add n back for σ(n).

σ(n) = s(n) + n
Intermediate: 28 + 28
Result: σ(28) = 56
Step 5

Compare s(n) against n.

s = n perfect · s < n deficient · s > n abundant
s(n):28
n:28
gap:0
Result: PERFECT
Step 6

Verify the classification.

σ(n) = 2n check
Intermediate: σ = 56 = 2×28
Result: Confirmed perfect: 1 + 2 + 4 + 7 + 14 = 28
Final Answer
perfect

Worked Examples

See how the calculator works with different scenarios.

Scenario

Smallest perfect number.

Inputs

n:6

Calculation

1+2+3 = 6

Result

PERFECT

Interpretation

Aliquot sum equals n exactly.

Understanding Your Result

Your Result: Classification — perfect, deficient, or abundant (see the result card with s(n) and σ(n)).

Typical range: Most numbers are deficient; abundant numbers start at 12; perfects are vanishingly rare (6, 28, 496, 8128, …).

What This Means

Perfect balances exactly; deficient falls short; abundant overflows — the gap s(n)−n tells which.

Why It Matters

Aliquot sums drive amicable-pair detection, sociable chains, and the 2000-year hunt for odd perfects.

Professional Interpretation

Perfect verdict at scale hints at Mersenne-prime structure; abundance measures divisor richness for packing problems.

Recommendations

  • •Quote divisors with the verdict
  • •Use σ(n)=2n as the perfect cross-check
  • •Report the gap, not just the label
  • •Cap loops at 10⁷
  • •Chain into amicable testing

Educational Content

Introduction

Perfect numbers balance exactly on their divisors — a 2000-year-old idea still generating open problems.

Concept Explanation

Pair divisors to √n, sum the proper ones, and compare the stack against n itself.

Applications

▸Amicable and sociable chains
▸Mersenne-prime hunting
▸Divisor-rich packing sizes
▸Classroom factor practice
▸History of Greek mathematics
▸Abundancy-index studies

Benefits

  • •Full aliquot list shown
  • •σ and s together
  • •Gap quantified
  • •Balance visual included
  • •Mersenne connection noted

Limitations

  • •Divisor loops to 10⁷
  • •Odd-perfect question unsettled
  • •Single n per run

Best Practices

  • •List every proper divisor
  • •Verify s(n) by re-adding
  • •State gap with sign
  • •Cross-check σ(n) = 2n for perfects
  • •Note Euclid–Euler for evens

Common Misconceptions

Myth: Perfect means "no divisors".
Truth: Perfect means divisors sum exactly to n — 28 has five of them.
Myth: Odd perfect numbers are known.
Truth: None is known; existence is a famous open problem.
Myth: All abundant numbers are multiples of 12.
Truth: 12 is just the first; 18, 20, 24 follow without that pattern.
Myth: 1 is perfect.
Truth: s(1)=0, so 1 is deficient.

Practical Tips

  • •List divisors in pairs to √n
  • •Always exclude n itself from s(n)
  • •Use σ(n) = 2n to double-check perfects
  • •Report the signed gap
  • •Remember 6, 28, 496, 8128
  • •Know 12 is the first abundant
  • •Cap inputs at 10,000,000

Common Mistakes

Including n in the aliquot sum
That computes σ(n); subtract n back out for s(n).
Missing the paired large divisor
Each small d gives n/d — collect both below √n.
Calling 1 perfect
1 has no proper divisors: s(1)=0 makes it deficient.
Confusing abundant with perfect
Perfect needs exact equality, not merely s(n) ≥ n.
Scanning all the way to n
Stop at √n with pairing — O(√n), not O(n).

Assumptions

This calculator makes the following assumptions to provide estimates:

Positive integer: n ≥ 1; fractions rejected.
Divisor-loop scope: n ≤ 10,000,000 for instant pairing.
Exact sums: Integer addition, no rounding.
Standard definitions: s(n)=σ(n)−n; perfect ⟺ equality.
Local only: No server calls.

Limitations

Be aware of these limitations when using this calculator:

Input cap: Above 10⁷ enumeration slows.
Single value: Ranges need repeated runs.
No odd-perfect proof: Numeric test only, not theory.
Chart sampling: Bars show at most the first entries.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a perfect number?
A positive integer equal to the sum of its proper divisors: s(n) = n. Examples: 6 = 1+2+3 and 28 = 1+2+4+7+14.
Is 28 perfect?
Yes — proper divisors 1, 2, 4, 7, 14 sum to 28 exactly.
What are deficient and abundant numbers?
Deficient: s(n) < n (e.g. 8 → 1+2+4 = 7). Abundant: s(n) > n (e.g. 12 → 1+2+3+4+6 = 16).
What is σ(n) vs s(n)?
σ(n) sums all divisors including n; s(n) = σ(n) − n sums proper divisors only. Perfect means σ(n) = 2n.
Are there odd perfect numbers?
Unknown — none found to 10^1500+, and proofs require ≥ 8 prime factors. An open problem.
How do even perfect numbers arise?
Euclid–Euler: every even perfect number is 2^(p−1)(2^p − 1) with 2^p − 1 a Mersenne prime (6, 28, 496, 8128, …).
Is 496 perfect?
Yes — the third perfect number: proper divisors sum to 496.
Why the 10,000,000 cap?
Divisor enumeration to √n stays instant below 10⁷.
Is 1 perfect, deficient, or abundant?
Deficient: it has no proper divisors, so s(1) = 0 < 1.
Are perfect numbers rare?
Extremely — only 51 are known (all even), the largest having tens of millions of digits.

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Related Guides

Deepen your understanding with these educational resources.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Perfect number
n with s(n) = n (e.g. 6, 28).
Deficient number
n with s(n) < n.
Abundant number
n with s(n) > n.
Aliquot sum s(n)
Sum of proper divisors.
Sigma σ(n)
Sum of all divisors including n.
Mersenne prime
Prime 2^p − 1 generating even perfects.
Euclid–Euler theorem
Characterizes all even perfect numbers.
Abundancy index
σ(n)/n ratio measuring divisor richness.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

Common Core 4.OA.B — factor pairs and multiplesNCTM Number and Operations — divisibility and classification

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

License:

Free to use for educational and professional purposes.