Calculate sigma(n), the sum of all positive divisors
Result
Answer
Step-by-Step Derivation
Divisor Sum Function Formula
If n = p1^a1 x p2^a2 x ... x pk^ak
sigma(n) = product of (p^(a+1) - 1) / (p - 1)
Proper divisor sum = sigma(n) - n
The divisor sum function adds every positive divisor of n. Prime factorization turns the divisor list into a product of geometric sums, one for each prime power.
⚠This calculator gives sigma(n), including n itself. For proper divisors, subtract n from the displayed sigma value.
What Is the Divisor Sum Function?
The divisor sum function, written sigma(n), measures the total of all positive divisors of a number. It connects factorization to perfect, abundant, and deficient number classification.
Includes n
Unlike proper divisor sum, sigma(n) includes the original number.
Prime Powers
Each p^a contributes a finite geometric sum.
Multiplicative
Coprime factors let divisor sums multiply.
Classification
Compare sigma(n)-n with n to classify numbers.
Example: For n = 36 = 2^2 x 3^2, sigma(36) = (2^3-1)/(2-1) x (3^3-1)/(3-1) = 7 x 13 = 91.
Applications of Divisor Sum Function
Perfect NumbersAbundant NumbersDivisor TheoryMultiplicative FunctionsContest Math
Frequently Asked Questions
What is a divisor sum function calculator?▼
It calculates sigma(n), the sum of all positive divisors of n, including 1 and n itself.
What is the sigma(n) formula?▼
If n equals the product of p^a terms, then sigma(n) is the product of (p^(a+1)-1)/(p-1) for each prime power.
How do I use this divisor sum calculator?▼
Enter a positive integer and calculate. The tool factors n and applies the divisor sum formula.
Is sigma(n) the same as proper divisor sum?▼
No. sigma(n) includes n itself, while the proper divisor sum is sigma(n)-n.
Where is the divisor sum function used?▼
It is used for perfect numbers, abundant and deficient numbers, divisor theory, multiplicative functions, and math contests.
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