Divisor Functions
Count and sum divisors.
Key idea
Count and sum divisors.
In depth
τ(n) counts divisors and σ(n) sums them. For n = Πpᵃ, τ = Π(a + 1) and σ = Π(pᵃ⁺¹ − 1)/(p − 1). They are used to define perfect numbers.
Worked example
n = 36 = 2²3²: τ = 9, σ = 7 × 13 = 91.
Formula
Real-world uses
- Counting factors of a number to analyse how many divisors an integer has in programming problems
Common mistake
Forgetting to include 1 and n itself in the divisor count.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Divisor Functions?
Count and sum divisors. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Divisor Functions?
The key formula is σ(n) = Σ d.
Where is Divisor Functions used in real life?
Real-world examples: Counting factors of a number to analyse how many divisors an integer has in programming problems.
What should I learn before Divisor Functions?
Make sure you are comfortable with Multiplicative Functions, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Divisor Functions?
Forgetting to include 1 and n itself in the divisor count.
How can I practise Divisor Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Divisor Functions?
Move on to Perfect and Amicable Numbers next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.