Advanced Divisor Function Calculator

Divisor Function Inputs

Positive integer n

Maximum accepted value is 1,000,000,000,000.

Power k for σ_k(n)

Use 0 for divisor count, 1 for divisor sum.

Divisor list type

Range start

Range end

Decimal places

Example Data Table

n	Prime factorization	τ(n)	σ(n)	Number type
28	2² × 7	6	56	Perfect
36	2² × 3²	9	91	Abundant
49	7²	3	57	Deficient
60	2² × 3 × 5	12	168	Abundant
97	97	2	98	Prime

Formula Used

Let n = p₁^a1 p₂^a2 ... p_r^ar be the prime factorization of n.

τ(n) = Π(a_i + 1). This gives the number of positive divisors.
σ_k(n) = Π(1 + p_i^k + p_i^2k + ... + p_i^a_ik).
For k = 1, σ(n) is the sum of all positive divisors.
Proper divisor sum = σ(n) - n.
Unitary divisor count = 2^ω(n), where ω(n) counts distinct prime factors.
φ(n) = n Π(1 - 1/p_i). It counts positive integers up to n that are coprime to n.
μ(n) is 0 if any prime exponent is above 1. Otherwise it is (-1)^ω(n).

How to Use This Calculator

Enter a positive integer in the n field.
Enter k to calculate the generalized divisor sum σ_k(n).
Choose which divisor list you want to display.
Set a range to compare nearby values in the chart.
Press the calculate button. The result appears above the form.
Use the CSV or PDF button to save the output.

Divisor Functions in Number Theory

A divisor function studies how numbers divide into another number. It turns a single integer into useful structural facts. The most common version is tau of n. It counts every positive divisor of n. Another version is sigma of n. It adds all positive divisors of n. A wider form, sigma k of n, adds each divisor raised to a chosen power.

Why These Values Matter

Divisor functions appear in number theory, algebra, cryptography, and contest mathematics. They help reveal if a number is prime, highly composite, abundant, deficient, or perfect. They also support modular arithmetic checks. A divisor count can show how many factor pairs exist. A divisor sum can compare a number with its proper factors. These ideas are simple, but they create deep patterns.

Prime Factorization Method

The calculator uses prime factorization because it is faster than testing every possible divisor. If n equals p one to a one times p two to a two, each exponent controls the divisor count. The count is the product of each exponent plus one. The sum formula uses a geometric series for every prime power. This avoids listing every divisor before computing the main result.

Advanced Calculator Features

This tool calculates tau n, sigma n, generalized sigma k, proper divisor sum, aliquot sum, unitary divisor data, Euler phi, Möbius value, and divisor product. It also creates a divisor list. The chart compares nearby values across a selected range. This makes patterns easier to see. You can export the visible result as CSV or PDF for reports and lessons.

Practical Use Cases

Students can check homework steps. Teachers can prepare examples. Developers can test factor based algorithms. Researchers can inspect special numbers quickly. Enter a number, select an exponent, choose a range, and calculate. The result card appears above the form. Use the divisor list for manual checking. Use the chart for trend comparison.

The definitions also connect with Dirichlet series and multiplicative functions. When two numbers are coprime, their divisor functions multiply cleanly. This property makes the formulas powerful. It also explains why prime factorization is the natural input for deeper divisor analysis in practice.

FAQs

1. What is a divisor function?

A divisor function returns information about the positive divisors of an integer. Common outputs include divisor count, divisor sum, and generalized sums using powers.

2. What does τ(n) mean?

τ(n) means the number of positive divisors of n. For example, 12 has divisors 1, 2, 3, 4, 6, and 12, so τ(12) equals 6.

3. What does σ(n) mean?

σ(n) is the sum of all positive divisors of n. It is the same as σ₁(n), because each divisor is raised to the first power.

4. What is σ_k(n)?

σ_k(n) adds each positive divisor after raising it to power k. When k is zero, the result becomes the divisor count.

5. What are proper divisors?

Proper divisors are positive divisors of n except n itself. Their sum helps classify numbers as perfect, abundant, or deficient.

6. What are unitary divisors?

A unitary divisor d divides n and has gcd(d, n/d) equal to 1. It uses full prime power blocks from the factorization.

7. Why use prime factorization?

Prime factorization makes divisor formulas fast. The calculator can compute counts and sums from exponents instead of testing every possible divisor first.

8. Can this calculator test prime numbers?

Yes. A number above 1 is prime when its factorization has one prime with exponent 1. In that case, τ(n) equals 2.