# Largest prime power divisor

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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

## Contents

## Definition

Let be a natural number. The **largest prime power divisor** of , sometimes denoted and sometimes denoted , is defined as the largest prime power that divides .

## Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A034699

### Upper bound

The value of is largest when itself is a prime power, namely, it is for these values of . Since there are infinitely many primes, we have:

.

### Lower bound

`Further information: Largest prime power divisor has logarithmic lower bound`

The largest prime power divisor of is . In fact, we have:

is finite and greater than zero.

Thus, we have:

.

### Asymptotic fraction

`Further information: Fractional distribution of largest prime power divisor`

The value of is almost uniformly distributed in the interval .

## Relation with other arithmetic functions

- Prime divisor count function: This is the total number of prime divisors of , and is denoted . We have the following relation:

.