# Largest prime power divisor

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 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:

.