---
title: "Divisors Calculator: List, Count and Sum | Prime Orchard"
description: "List every divisor of a number, count them and add them up, then see whether it is perfect, abundant or deficient, with formulas from its prime factors."
url: https://primeorchard.top/divisors-calculator/
---

Divisors

# Divisors calculator

All the divisors of a number, how many there are and what they add up to, and whether the number is perfect, abundant or deficient.

Divisors 6

Sum σ(n) 56

Type perfect

Its proper divisors (all except 28 itself) add up to 28, and 28 = 28.

### All 6 divisors

1. 1
2. 2
3. 4
4. 7
5. 14
6. 28

### Divisor pairs

- 1 × 28
- 2 × 14
- 4 × 7

### From the prime factorization

28 = 22 × 7

- Count: τ(n) = (2 + 1) × (1 + 1) = 6
- Sum: σ(n) = (23 − 1)/(2 − 1) × (1 + 7) = 56

## Divisors from prime factors

A divisor of n is a whole number that divides it with nothing left over; 12 has six: 1, 2, 3, 4, 6 and 12. Divisors come in pairs, d × (n ÷ d) = n, so a hand search only needs to go up to √n. Perfect squares are the only numbers with an odd count, because their square root pairs with itself.

For bigger numbers the calculator starts from the prime factorization. If n = pa × qb × …, every divisor chooses an exponent from 0 to a for p, from 0 to b for q, and so on. That gives the count and the sum directly:

τ(n) = (a + 1)(b + 1)… σ(n) = (1 + p + … + pa)(1 + q + … + qb)…

For 360 = 23 × 32 × 5 the count is 4 × 3 × 2 = 24 divisors and the sum is 15 × 13 × 6 = 1,170.

Now add the divisors except n itself and compare with n. Equal makes n **perfect** (6 = 1 + 2 + 3). More makes it **abundant** (for 12, 1 + 2 + 3 + 4 + 6 = 16). Less makes it **deficient**, as every prime is, since its only smaller divisor is 1.

## Sources

- OEIS sequences [A000005](https://oeis.org/A000005) (number of divisors), [A000203](https://oeis.org/A000203) (sum of divisors), [A000396](https://oeis.org/A000396) (perfect numbers) and [A005101](https://oeis.org/A005101) (abundant numbers).
- Euclid, *Elements*, Book IX, [Proposition 36](https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html): 2p−1(2p − 1) is perfect when 2p − 1 is prime; Euler later showed every even perfect number has this form.
- Ramanujan S. Highly composite numbers. *Proceedings of the London Mathematical Society* 14 (1915), 347–409.

## Questions

**Are there odd perfect numbers?**

Nobody knows. Every known perfect number is even and of the form 2p−1(2p − 1) with 2p − 1 prime, so there are exactly as many known perfect numbers as known Mersenne primes.

**What is the smallest odd abundant number?**

945 = 33 × 5 × 7, whose proper divisors add up to 975. Even abundant numbers are common, since every multiple of 6 above 6 is abundant; odd ones are much rarer.

**Which number below 1,000 has the most divisors?**

840, with 32. Numbers that have more divisors than every smaller number are called highly composite; Ramanujan studied them in 1915.