Abundant Number Calculator: Classifying Numbers by Their Divisor Sums
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
Every whole number falls into one of three buckets, and an abundant number calculator sorts them with a single comparison: add up everything that divides n evenly except n itself, then see whether that total beats n, falls short of it, or lands on it exactly. Twelve beats it — 1 + 2 + 3 + 4 + 6 = 16, so 12 is abundant by 4. Ten falls short — 1 + 2 + 5 = 8, deficient by 2. Six ties at 1 + 2 + 3 = 6, which is what makes it perfect. Three buckets, one subtraction, no ambiguity.
The tool above does more than announce the bucket. It builds σ(n) from the prime factorisation instead of grinding through a divisor list, prints the abundance and the abundancy index σ(n)/n side by side, and tells you whether the surplus can actually be packed back into n — because sometimes it cannot, and those numbers have a name. Below: why roughly one number in four is abundant, why the first odd one hides all the way out at 945, and the one arithmetic slip that turns every perfect number into an abundant one.
Reading the Scale: Abundant, Deficient, Perfect
Two quantities do all the work here and they are easy to mix up. The divisor sum σ(n) counts every divisor including n. The aliquot sum s(n) leaves n out, so s(n) = σ(n) − n. Classification always uses the aliquot sum. The gap between them, A(n) = s(n) − n, is the abundance — positive for abundant numbers, negative for deficient ones, zero for perfect ones.
There is a third way to read the same information, and it is the one that scales. The abundancy index σ(n)/n compares the divisor sum to the number as a ratio rather than a difference, and the cut-off is always 2 — no matter how large n is. That matters because raw abundance grows with n while the index does not. 100 is abundant by 17; 1,000,000 is abundant by 480,437. The surplus grew by a factor of 28,000, yet the index only moved from 2.17 to 2.48 — the two numbers are far more alike than the raw gap suggests, and the index is what says so.
| n | s(n) | A(n) = s(n) − n | σ(n)/n | Verdict |
|---|---|---|---|---|
| 12 | 16 | +4 | 2.333 | Abundant — the smallest |
| 28 | 28 | 0 | 2.000 | Perfect |
| 70 | 74 | +4 | 2.057 | Abundant, and weird |
| 97 | 1 | −96 | 1.010 | Deficient — prime |
| 945 | 975 | +30 | 2.032 | Abundant — first odd one |
| 1024 | 1023 | −1 | 1.999 | Deficient by exactly 1 |
That last row is worth a second look. Powers of two are always deficient by exactly 1, because 1 + 2 + 4 + … + 2k−1 = 2k − 1 for every k. They creep toward perfection forever and never arrive — which is precisely why perfect numbers need a Mersenne prime multiplied in to close the gap.
How an Abundant Number Calculator Gets σ(n) Without a Divisor List
Listing divisors works fine for 12. It stops being reasonable around six digits. The fix is that σ is multiplicative: factor n into prime powers, take one bracket per prime, multiply. For a prime power pa the bracket is just 1 + p + p2 + … + pa.
Take 360 = 23 × 32 × 5. The brackets are (1 + 2 + 4 + 8) = 15, (1 + 3 + 9) = 13, and (1 + 5) = 6. Multiply: 15 × 13 × 6 = 1,170. Subtract 360 and you have s(360) = 810, so 360 is abundant by 450 with an index of 3.25. Three small sums and two multiplications replaced writing out all 24 divisors and adding 23 of them.
The saving compounds. 963,761,198,400 is the smallest number with 6,720 divisors — you would be at it all day by hand. Its factorisation is 26 × 34 × 52 × 7 × 11 × 13 × 17 × 19 × 23, so σ is nine brackets multiplied together, and that is a two-minute job. Any prime factorization calculator gets you the input for it; the calculator above shows the filled-in brackets for whatever you type so you can check your own arithmetic against it.
One consequence falls straight out of the formula. For a prime p the only bracket is (1 + p), so σ(p) = p + 1 and s(p) = 1. Primes are as deficient as a number can get for its size, which is why long runs of them in a scan produce nothing abundant at all — something you can watch happen in the range scanner, or verify one number at a time with the prime calculator.
Roughly One Number in Four Is Abundant
Most people expect abundance to be a curiosity. It is not. Scan the first hundred integers and 22 of them are abundant. Push further and the share stabilises fast:
| Range | Abundant | Share |
|---|---|---|
| 1–100 | 22 | 22.00% |
| 1–1,000 | 246 | 24.60% |
| 1–10,000 | 2,488 | 24.88% |
| 1–100,000 | 24,795 | 24.80% |
| 1–200,000 | 49,481 | 24.74% |
The limit is real, not just an appearance — the natural density of the abundant numbers is known to exist and sits at roughly 0.2476, pinned between 0.2474 and 0.2480 by explicit computation. Deficient numbers take about 75.2%, and the perfect numbers are so sparse they round to zero: only four of them live below a million.
Why so many? Because abundance is contagious. If d is abundant or perfect and d divides m, then m is abundant too. Every multiple of 6 above 6 is therefore abundant, and that alone hands you a sixth of all integers. Add the multiples of 20, of 28, of 88, and of every other starter, and a quarter of the number line is an easy total to reach. This is also what makes the classification useful in factor-based work: knowing one small abundant divisor settles the question for infinitely many numbers at once.
The contagion has a striking endpoint. Every integer greater than 20,161 can be written as the sum of two abundant numbers — 20,161 is the last one that cannot. Abundance is common enough that past that threshold, the pairs never run out.
Why the First Odd Abundant Number Is 945
Drop the even numbers and abundance nearly vanishes. Below 100,000 there are 24,795 abundant numbers and only 210 of them are odd — under half a percent of the odd numbers in that range. The reason is a budget argument, and it is easier than it looks.
The abundancy index of a product of prime powers is the product of the individual indices, and σ(pa)/pa can never exceed p/(p − 1) however large you make the exponent. So each prime brings a hard ceiling: 2 contributes at most 2, 3 at most 1.5, 5 at most 1.25, 7 at most 1.167, 11 at most 1.1. An even number gets most of the way to the target of 2 from the factor of 2 alone. An odd number has to build the same total out of parts that are all barely above 1.
Multiply the ceilings for 3, 5 and 7 and you get 1.5 × 1.25 × 1.167 = 2.1875 — enough in principle, but only with unlimited exponents. The actual smallest candidate has to work harder: 105 = 3 × 5 × 7 gives (4/3)(6/5)(8/7) = 1.83, short of the mark. Bumping 3 to 33 turns its bracket into 40/27 = 1.481, and 945 = 33 × 5 × 7 finally clears the bar at 1.481 × 1.2 × 1.143 = 2.032. That is the whole reason the first odd abundant number sits at 945 rather than somewhere in the double digits.
70 Is Abundant and Cannot Spend It
Here is a question the classification does not answer. A number is abundant because its divisors overshoot it — so can you pick out some of those divisors that add up to exactly n, and leave the surplus behind? Usually yes. 12 does it with 6 + 4 + 2. A number that manages this is called semiperfect, and the vast majority of abundant numbers are.
70 is not, and the proof fits in a paragraph. Its proper divisors are 1, 2, 5, 7, 10, 14 and 35, totalling 74. Either 35 is in your subset or it is not. If it is not, the largest total available is 1 + 2 + 5 + 7 + 10 + 14 = 39, nowhere near 70. If it is, you need 35 more from a set that totals 39 — meaning you must leave out exactly 4, and no subset of {1, 2, 5, 7, 10, 14} sums to 4. Both branches fail, so no subset works. 70 is the smallest weird number: abundant, yet unable to spend its abundance.
Weird numbers are genuinely rare — 70, 836, 4,030, 5,830, 7,192, 7,912 and 9,272 are all of them below 10,000. Nobody has ever found an odd one, and none exists below 1021, but no proof rules them out either. That is why the calculator above runs a real subset-sum rather than a shortcut: greedy divisor-picking succeeds for almost everything, and the cases where it fails are exactly the interesting ones.
A related label sorts the abundant numbers by where they come from. A primitive abundant number is abundant while every one of its proper divisors is deficient — it starts a family rather than inheriting membership. The first few are 20, 70, 88, 104, 272 and 304; the smallest odd one is 945 again. Since every abundant number is a multiple of some primitive abundant number, that short list generates the whole quarter of the number line.
The Slip That Turns Every Perfect Number Abundant
One mistake accounts for more wrong answers here than everything else combined: including n in its own divisor sum. Do that to 6 and you get 1 + 2 + 3 + 6 = 12, which looks abundant. Do it to any perfect number and you get exactly 2n, which also looks abundant. The classification runs on s(n), never σ(n), and the two differ by precisely n. If your results say every number you test is abundant, this is why.
Three more traps worth naming:
- Forgetting that 1 counts. 1 divides everything, and dropping it costs you a full point of aliquot sum. For 12 that turns 16 into 15 and flips the verdict to deficient.
- Double-counting the square root. Pairing divisors as d × (n/d) is the fast way to list them, but for perfect squares the middle pair is the same number twice. 36 has divisor 6 paired with itself; count it once, or s(36) reads 61 instead of 55.
- Assuming lots of divisors means abundant. Divisor count and divisor sum are different things. 16 has five divisors and is deficient; 12 has six and is abundant. Small divisors barely move the sum — what matters is how much of n the large ones cover.
Everything on this page runs on the same object, the divisor structure of n. Once you have it, the abundant/deficient/perfect verdict is one subtraction, the Euler totient φ(n) is one more product over the same primes, and tests like the Armstrong number check come from the digits instead. For the definitions, counts and open problems around these classes, the Wikipedia entry on abundant numbers is a reliable starting point.
