Prime divisor count function

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).
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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:

.