Deficient Number Calculator: The Third Class Most Lists Leave Out
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
Roughly three out of every four whole numbers are deficient, and a deficient number calculator tells you in one subtraction whether yours is one of them. Add up a number’s proper divisors — every divisor except the number itself — and compare the total to the number. Falls short? Deficient. Matches exactly? Perfect. Overshoots? Abundant. Perfect and abundant numbers get all the attention, which is odd, because they’re the rare exceptions. This page covers the majority class: how the check works, why entire families of numbers can never escape it, and the one open question about deficiency that’s been unsolved for centuries.
How a Deficient Number Calculator Decides in One Subtraction
The quantity doing the work is the aliquot sum, written s(n): the sum of all proper divisors of n. Take 21. Its divisors are 1, 3, 7 and 21; drop the number itself and the proper divisors are 1, 3 and 7. So s(21) = 11, which is well below 21 — deficient, with a deficiency of 21 − 11 = 10.
There’s a shortcut worth knowing. The divisor function σ(n) sums all divisors including n, and it has a clean product formula built from the prime factorization. The aliquot sum is then just s(n) = σ(n) − n. That’s how this calculator handles numbers up to a trillion without listing anything: it factors the number (you can do the same step separately with the prime factorization calculator), applies the formula, and subtracts. For 21: σ(21) = (1 + 3)(1 + 7) = 32, and 32 − 21 = 11. Same answer, no list needed.
Check 16, 18 and 19 by Hand
Three consecutive-ish numbers, three different stories. Working them by hand shows how sensitive the classification is:
| n | Proper divisors | s(n) | Verdict |
|---|---|---|---|
| 16 | 1, 2, 4, 8 | 15 | Deficient by exactly 1 |
| 18 | 1, 2, 3, 6, 9 | 21 | Abundant by 3 |
| 19 | 1 | 1 | Deficient by 18 |
Notice the spread. 19 misses by 18 — about as deficient as a number that size can be. 16 misses by the smallest possible margin. And 18, sitting between them, tips over into abundance because it stacks two small prime factors (2 and 3²) that generate lots of mid-sized divisors. Divisor wealth isn’t about being big; it’s about being built from small primes. If the abundant side of the line is what you’re after, the abundant number calculator digs into that half of the story.
Why Is Every Prime Power Deficient?
A prime p has one proper divisor: 1. So s(p) = 1, and no prime will ever be anything but deficient. The surprise is that raising a prime to any power doesn’t help. The proper divisors of pk are 1, p, p², …, pk−1 — a geometric series that sums to (pk − 1)/(p − 1). For p = 2 that’s 2k − 1, one short of the number. For any larger prime the shortfall only grows: the series sums to less than half of pk when p ≥ 3.
| Prime power | s(n) | Deficiency |
|---|---|---|
| 8 = 2³ | 1 + 2 + 4 = 7 | 1 |
| 27 = 3³ | 1 + 3 + 9 = 13 | 14 |
| 125 = 5³ | 1 + 5 + 25 = 31 | 94 |
The pattern in that third column matters: powers of 2 always miss by exactly 1, while powers of odd primes miss by more than half the number. Escaping deficiency requires combining different primes — and even that usually isn’t enough. The product of two distinct odd primes is always deficient too: s(3 × 5) = 1 + 3 + 5 = 9 < 15. In fact every odd number below 945 is deficient. Not one exception.
Deficiency 1: The Almost Perfect Club
Numbers whose aliquot sum lands exactly one short — s(n) = n − 1 — are called almost perfect. Every power of 2 qualifies: 16 has s(16) = 15, 32 has s(32) = 31, and so on forever, since 1 + 2 + … + 2k−1 = 2k − 1. Here’s the part that should bother you: nobody has ever found an almost perfect number that isn’t a power of 2, and nobody has proved none exists. The question has resisted proof for centuries, which puts it in the same family of stubborn problems as the search for odd perfect numbers. Type a few powers of 2 into the calculator above and watch the deficiency pin to 1 every single time — then try to find anything else that does it. Sequence A005100 in the OEIS catalogs the deficient numbers if you want to explore the full list.
How Rare Is a Non-Deficient Number?
Rarer than most people guess. Run the range scanner above at a few different limits and the split barely moves:
| Up to | Deficient | Perfect | Abundant | % deficient |
|---|---|---|---|---|
| 100 | 76 | 2 | 22 | 76.0% |
| 1,000 | 751 | 3 | 246 | 75.1% |
| 10,000 | 7,508 | 4 | 2,488 | 75.1% |
| 1,000,000 | 752,451 | 4 | 247,545 | 75.2% |
That stability is real, not a coincidence of small samples — the density of deficient numbers converges to roughly 75.2% of all integers. Perfect numbers, meanwhile, are absurdly scarce: four below a million, and only around 50 known in total even with modern computer searches. So when a textbook presents “deficient, perfect, abundant” as three parallel categories, keep the actual proportions in mind. It’s a landslide, a rounding error, and a solid minority.
Deficient Numbers Run in the Family
One property makes deficient numbers structurally different from abundant ones: deficiency is inherited downward. Every proper divisor of a perfect number is deficient, and every proper divisor of a deficient number is deficient too. Check it against 28, which is perfect: its proper divisors 1, 2, 4, 7 and 14 are all deficient (s(14) = 1 + 2 + 7 = 10, for instance). Abundance works in the opposite direction — every multiple of an abundant number is abundant, which is why once 12 shows up, 24, 36, 48 and the rest of its multiples are locked in.
This gives you a fast mental filter. A number that divides a known perfect or deficient number never needs checking. And a number divisible by 12, 18, 20 or any other known abundant number is abundant, full stop. The hand computation is only ever needed for the cases the two inheritance rules don’t already settle.
Where the Hand Check Goes Wrong
Two specific slips account for nearly every wrong classification, and both are worth naming because each one flips the verdict rather than just nudging a digit:
- Counting the number as its own divisor. Include 16 in the divisor sum and you get 31, which reads as abundant. The correct proper-divisor sum is 15 — deficient. This single mistake converts every deficient number with s(n) > something small into a false abundant, because adding n to the sum always pushes the total past n.
- Forgetting 1. Every number greater than 1 has 1 as a proper divisor. Drop it and 16’s sum becomes 14 instead of 15 — still deficient here, but the same slip turns the perfect number 6 (1 + 2 + 3 = 6) into a false deficient (2 + 3 = 5).
One more edge case trips people up: the number 1. It has no proper divisors at all, so s(1) = 0, which is less than 1 — making 1 the smallest deficient number. Some older texts exclude it by convention, but under the standard definition it counts, and this calculator classifies it that way.
