Sum of Divisors Calculator
The sum of divisors of a whole number, written sigma of n, adds up every positive integer that divides it evenly, including one and the number itself. This calculator takes a positive whole number and returns that total. It computes sigma efficiently from the prime factorization: for each prime power in the factorization it adds up the geometric series of that prime's powers, then multiplies the results across all primes. For sixty, which factors as two squared times three times five, sigma is the product of one plus two plus four, one plus three, and one plus five, giving seven times four times six, which is one hundred sixty-eight. The sigma function is central to number theory and underpins perfect numbers, where sigma equals twice the number, and abundant or deficient numbers. The calculator also lists the divisors so you can verify the sum. Mathematicians, students and puzzle solvers use it to study the multiplicative structure of integers. Enter a whole number to get the sum immediately; the input is a plain integer, so no decimal formatting applies. Every figure here is computed deterministically from the rule shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
Sigma adds every divisor of n. For 60, sigma equals (1+2+4)(1+3)(1+5) = 7 x 4 x 6 = 168.
Sum of Divisors formula
sigma(n) = product over primes of (p^(e+1) - 1)/(p - 1)
n = p1^e1 x p2^e2 x ... x pk^ek
each factor sums 1 + p + p^2 + ... + p^e
sigma(n) = sum of all positive divisors
includes 1 and n itself
For each prime power in the factorization, add the geometric series of that prime's powers from one up to the full exponent, then multiply those sums across all primes.
Worked example
Find the sum of divisors of 60.
- Factor: 60 = 2^2 x 3 x 5
- Prime 2: 1 + 2 + 4 = 7; prime 3: 1 + 3 = 4; prime 5: 1 + 5 = 6
- sigma = 7 x 4 x 6 = 168
The sum of divisors of 60 is 168. This is the calculator's default input, so the result above matches the widget exactly.
Sigma for sample numbers
sigma(n) sums all divisors of n.
| n | Factorization | Sigma |
|---|---|---|
| 6 | 2 x 3 | 12 |
| 12 | 2^2 x 3 | 28 |
| 28 | 2^2 x 7 | 56 |
| 60 | 2^2 x 3 x 5 | 168 |
| 100 | 2^2 x 5^2 | 217 |
Number-theory reference: US National Institute of Standards and Technology (NIST).
Sum of Divisors Calculator: frequently asked questions
What is the sum of divisors?
It is the total you get by adding every positive integer that divides a number evenly, including one and the number itself. For 6 the divisors are 1, 2, 3 and 6, so the sum, sigma of 6, is 12. The function is usually written with the Greek letter sigma.
Why use the prime factorization?
Because sigma is multiplicative over coprime parts, you can compute it prime by prime. For each prime power you sum the geometric series of its powers, then multiply those partial sums. This is far faster than listing and adding every divisor for large numbers.
How does sigma define a perfect number?
A perfect number is one whose divisors, excluding itself, sum to the number, which is the same as saying sigma of n equals twice n. For example sigma of 28 is 56, exactly twice 28, so 28 is perfect.
What are abundant and deficient numbers?
If sigma of n is more than twice n the number is abundant, and if it is less the number is deficient. Most numbers are deficient; abundant numbers like 12 and 24 have divisors that more than account for the number itself.
What is the sigma formula?
For each prime power in the factorization, sum one plus the prime plus its higher powers up to the exponent, then multiply across primes. For 60 that is (1+2+4)(1+3)(1+5), which is 7 times 4 times 6, equal to 168.
Official sources
- Mathematical functions and integer-sequence reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.