# Prime divisor count function

From Number

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).

View a complete list of arithmetic functions

## Definition

Let be a natural number. The **prime divisor count function** of , denoted , is defined as the number of prime divisors of .

## Relation with other arithmetic functions

- Mobius function: This is defined as for a square-free number, and is otherwise.
- Divisor count function: This is denoted , and is defined as the total number of positive divisors of .
- Largest prime power divisor: Denoted , this is defined as the largest prime power dividing . We have:

.