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:
.
