Definition
s(n) = n
Proper divisor sum equals the number.
Check Perfect Number in Python
Overview
What You’ll Learn
A perfect number equals the sum of its proper divisors. For example, 6 is perfect because 1 + 2 + 3 = 6. This tutorial covers the definition, deficient/abundant neighbors, a live checker, worked Python examples, edge cases, and complexity.
Proper Divisors
Exclude n
Positive divisors of n except n itself.
Scan to n//2
Simple loop
No proper divisor exceeds n // 2.
Classify
s(n) vs n
Deficient, perfect, or abundant.
Live Preview
Try 28 / 12
See divisors, sum, and verdict live.
Rare
6, 28, 496…
Perfect numbers are uncommon.
Introduction
A perfect number is a positive integer whose proper divisors add up to the number itself. Using s(n) for that sum: perfect means s(n) = n, deficient means s(n) < n, and abundant means s(n) > n.
Equivalently, if sigma(n) is the sum of all positive divisors, then perfect means sigma(n) = 2n. One is never perfect: by convention s(1) = 0.
Why it matters?
It is a classic divisor-sum interview problem that connects loops, modulo, and number-theory vocabulary.
Key Highlights
s(n) = n
Proper divisors sum to n.
Exclude n
Do not add the number itself.
1 Is Not
s(1) = 0 by convention.
Class Neighbors
Deficient / abundant too.
In short: sum divisors from 1 to n // 2; perfect when that sum equals n.
Foundation
📝 Problem & Approach
Given a positive integer n, decide whether it is perfect by comparing the sum of its proper divisors with n.
python
# 6: 1 + 2 + 3 = 6 -> perfect
# 28: 1+2+4+7+14 = 28 -> perfect
# 10: 1 + 2 + 5 = 8 -> deficient
# 12: 1+2+3+4+6 = 16 -> abundantInputs & Outputs
| Item | Type | Description |
|---|---|---|
n / number | int | Positive integer to classify. |
| Return | bool | True when proper divisor sum equals n. |
| Classification | text | Deficient, perfect, or abundant. |
Minimal workflow
Pseudocode
function proper_divisor_sum(n):
sum = 0
for i from 1 to floor(n / 2):
if n mod i == 0:
sum = sum + i
return sum
function is_perfect(n):
if n < 2:
return false
return proper_divisor_sum(n) == nMethod comparison
| Method | Idea | Notes |
|---|---|---|
| Scan to n // 2 | Add every proper divisor | Interview default — clearest |
| Pair to sqrt(n) | Add i and n // i | Faster; careful with squares |
| sigma(n) = 2n | All-divisor sum | Equivalent definition |
Cheat Sheet
⚡ Quick Reference
| Goal | Pattern |
|---|---|
| Loop bound | range(1, n // 2 + 1) |
| Is divisor? | if n % i == 0: total += i |
| Perfect? | return total == n |
| Deficient | total < n |
| Abundant | total > n |
| Guard 1 | if n < 2: return False |
Compare
📋 Perfect vs Deficient vs Abundant
Same divisor sum — different comparisons to n.
Perfect
s(n) = nThis page — e.g. 6, 28
Deficient
s(n) < nMost numbers, including primes
Abundant
s(n) > ne.g. 12 — related topic
Interview tip
exclude nProper divisors only
Context
When This Problem Shows Up
Reach for a proper-divisor sum whenever you need to classify perfection.
Number-theory warm-ups
Divisors, sums, and classification.
Modulo practice
Find all divisors with %.
Bridge to abundant
Same sum, different comparison.
Range hunting
Find the rare perfect values in a band.
Not for huge scans
Naive O(n) per check gets costly fast.
Key benefit: one clear loop that teaches proper divisors, classification, and the famous examples 6 and 28.
Interactive
🔮 Live Preview
Enter a positive integer and inspect proper divisors, sum, and verdict.
Hands-On
Examples Gallery
Three complete Python programs — check 28, list perfect numbers from 1 to 50, and classify deficient/perfect/abundant. Click View Output to reveal sample console results.
📚 Getting Started
A reusable helper and the classic example 28.
Example 1 — Check One Number
Test a fixed value (28) with a helper function.
python
def is_perfect_number(number: int) -> bool:
total = 0
for i in range(1, number // 2 + 1):
if number % i == 0:
total += i
return total == number
number = 28
if is_perfect_number(number):
print(f"{number} is a perfect number.")
else:
print(f"{number} is not a perfect number.")How It Works
The loop adds every proper divisor of 28: 1, 2, 4, 7, and 14. Their sum is 28, so the helper returns True.
⚡ Hunting in a Range
Reuse the helper to find the rare perfect values nearby.
Example 2 — Perfect Numbers in Range 1 to 50
Print all perfect numbers in a small interval.
python
def is_perfect_number(num: int) -> bool:
total = 0
for i in range(1, num // 2 + 1):
if num % i == 0:
total += i
return total == num
print("Perfect Numbers in the range 1 to 50:")
for i in range(1, 51):
if is_perfect_number(i):
print(i, end=" ")How It Works
Within 1..50 only 6 and 28 are perfect. That rarity is typical — the next known values jump much larger.
Example 3 — Classify Deficient / Perfect / Abundant
Reuse the same sum to label each sample number.
python
def proper_divisor_sum(n: int) -> int:
if n < 2:
return 0
total = 0
for i in range(1, n // 2 + 1):
if n % i == 0:
total += i
return total
def classify(n: int) -> str:
s = proper_divisor_sum(n)
if s == n:
return "perfect"
if s < n:
return "deficient"
return "abundant"
for value in (6, 10, 12, 28, 1):
print(f"{value}: {classify(value)} (s={proper_divisor_sum(value)})")How It Works
One sum drives three labels. Perfect is the equality case; deficient and abundant are the strict inequalities.
🧠 How the Algorithm Decides
1
Start sum at 0
Accumulate proper divisors only.
Init
2
Scan 1 .. n//2
Add i whenever n % i == 0.
Loop
3
Compare to n
Equal means perfect; else deficient/abundant.
Rule
=
Return the verdict
Bool for perfect, or a class label.
Trace
🔎 Worked Walkthrough — 28
Trace the proper-divisor sum for n = 28.
| i | 28 % i | Add? | total |
|---|---|---|---|
1 | 0 | Yes | 1 |
2 | 0 | Yes | 3 |
4 | 0 | Yes | 7 |
7 | 0 | Yes | 14 |
14 | 0 | Yes | 28 |
total 28 equals n — perfect.
Use Cases
Where perfect-number checks show up beyond the interview prompt.
1. Interview Classics
Divisor loops and equality checks.
Example: is_perfect(28).
2. Classification Sets
Label deficient / perfect / abundant.
Example: Example 3.
3. Range Searches
Find rare perfect values in a band.
Example: 6 and 28 in 1..50.
4. Teaching Divisors
Show what “proper” excludes.
Example: do not add n.
5. Bridge to Abundant
Same sum, greater-than test.
Example: related topic.
6. Next: Perfect Square
Another “perfect” naming cousin.
Example: related CTA.
Pro Tip: open with “proper divisors exclude n; perfect means sum equals n” before coding.
Advantages
Why the n//2 scan works well for beginners and interviews.
1. Easy to Trace
Dry-run 6 or 28 on paper and watch the sum grow.
2. Clear Bound
Stopping at n // 2 avoids adding n by mistake.
3. Extends to Classes
Same sum powers deficient/abundant labels.
4. Speeds Up Later
You can upgrade to sqrt pairing when needed.
Pro Tip: lead with the n//2 scan; mention sqrt pairing only as an optimization aside.
Usage Tips
Small habits that keep perfect-number solutions interview-ready.
1. Exclude n Itself
Proper divisors stop at n // 2.
2. Guard Small n
Treat n < 2 as not perfect.
3. Name the Classes
Mention deficient and abundant alongside perfect.
4. Know 6 and 28
Use them as quick sanity checks.
5. Optimize Later
sqrt pairing is optional after the clear O(n) version.
Pro Tip: dry-run 6, 10, and 12 — if those three classes match, your sum logic is correct.
Common Pitfalls
Mistakes that commonly break perfect-number programs.
1. Including n in the Sum
Adding the number itself doubles the definition.
→ Stop at n // 2.
2. Calling 1 Perfect
Thinking 1 divides itself uniquely.
→ s(1) = 0; 1 is not perfect.
3. Confusing With Perfect Square
Different “perfect” concept entirely.
→ This page is about divisor sums.
4. Off-by-One Bound
Using range(1, n // 2) and missing n // 2.
→ Use range(1, n // 2 + 1).
5. Huge Naive Scans
Checking every n up to millions with O(n) each.
→ Use smaller ranges or faster pairing.
Edge Cases
Handle these before claiming the check is complete.
n = 1
Not perfect
Proper divisor sum is 0, not 1.
Prime
Always deficient
Only proper divisor is 1, so sum < n.
Exclude n
Proper divisors only
Adding n itself breaks the definition.
6 / 28
Classic yes cases
Use them as sanity checks.
12
Abundant sample
s(12) = 16 > 12.
Odd?
Open problem
No odd perfect number is known.
Theory
⚖️ Facts Worth Knowing
Handy follow-ups interviewers sometimes ask.
- First four. 6, 28, 496, and 8128.
- sigma form. Perfect ⇔ sigma(n) = 2n.
- Even via Mersenne. Even perfect numbers relate to Mersenne primes.
- Odd unknown. No odd perfect number has been found.
Drill
🎯 Practice Problems
Try these variations to lock in the pattern.
1. Prove 28
- List proper divisors
- Confirm sum = 28
2. Find 6 and 28
- Reproduce Example 2
- No other hits in 1..50
3. Classify 10 and 12
- Deficient vs abundant
- Match Example 3
4. Reject 1
- Show s(1) = 0
- Assert not perfect
Important
Notes
- Definition: perfect means proper divisor sum equals n.
- Loop: check divisors from 1 to n // 2 with modulo.
- Edge case: 1 is not perfect.
- You can test divisors only up to sqrt(n) and add divisor pairs. Perfect numbers are rare, so broad brute-force scans are expensive.
Quick Takeaway: sum proper divisors to n // 2; perfect when that sum equals n.
Analysis
⏱️ Time and Space Complexity
| Task | Time | Extra space |
|---|---|---|
| Single check with n // 2 scan | O(n) | O(1) |
| Single check with sqrt(n) pairing | O(sqrt(n)) | O(1) |
| Range scan 1..U (naive) | O(U²) | O(1) |
For interview demos, the O(n) scan is fine; mention pairing when asked about speed.
🎉 Conclusion
A perfect number equals the sum of its proper divisors. Loop from 1 to n // 2, add every divisor, and compare with n — remembering that 1 is not perfect and that perfect values are rare.
Practice the three examples above, then continue to checking perfect squares.
s(n) = n means perfect; exclude n from the divisor sum.
Pro Tips
💡 Best Practices
✅ Do
- Exclude n from the sum
- Loop to n // 2 inclusive
- Reject n < 2 early
- Sanity-check with 6 and 28
- Name deficient / abundant too
❌ Don’t
- Add n into the divisor sum
- Call 1 perfect
- Confuse with perfect squares
- Miss the +1 in range end
- Brute-force huge ranges blindly
Summary
Key Takeaways
Knowledge Unlocked
Five things to remember about perfect numbers
Classify divisor sums the interview-friendly way.
5
Core concepts 01
Rule
s(n) = n
Definition 02
Bound
to n // 2
Loop 03
Edge
1 not perfect
Guard 04
Classes
def / abun
Neighbors 05
Cost
O(n) naive
Analysis❓ Frequently Asked Questions
A positive integer is perfect if the sum of its proper divisors equals the number itself.
No. Its proper divisor sum is treated as 0, not 1.
If proper divisor sum is less than n it is deficient, if greater than n it is abundant.
No proper divisor of n can be larger than n // 2.
Yes. It is mathematically equivalent to proper-divisor sum equals n.
No, they are very rare.
Positive divisors of n excluding n itself. For 6 they are 1, 2, and 3.
Scan only up to sqrt(n) and add divisor pairs, being careful not to double-count squares.
Yes. 1 + 2 + 4 + 7 + 14 = 28.
Did you Know? 🔊
The first four perfect numbers are 6, 28, 496, and 8128. Whether any odd perfect number exists is still unknown.
Continue to Perfect Square
Learn how to check whether a number is a perfect square in Python.
About the author
9 people found this page helpful