# Arithmetic function

[https://en.wikipedia.org/wiki/Arithmetic\_function](https://en.wikipedia.org/wiki/Arithmetic_function)

> An **arithmetic function** or **number-theoretic function** is any function whose domain is the set ℤᐩ (positive integers) and whose range is a subset of ℂ (the complex numbers).

f

ᴀʙᴄᴅᴇꜰɢʜɪᴊᴋʟᴍɴᴏᴘǫʀsᴛᴜᴠᴡxʏᴢ

$$f: \mathbb{Z^+} \to S \ \text{where} \ S \subseteq \mathbb{C}$$

An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

```
divisors(n) = D(n) = { x | ∀x ∈ ℕ. x∣n ∧ 0<=x<=n }

divisors(28) = D(28) = { x | ∀x ∈ ℕ. x∣28 ∧ 0<=x<=n } = {1,2,4,7,14,28}

d(n) = |D(n)|
d(28) = |D(28)| = 6

divisor: n ⟼ d(n)
divisor(n) = d(n)

divisor(28) = 6

divisor(1) = 1
divisor(2) = 2
divisor(3) = 2
divisor(4) = 3
```

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