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).
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Definition

Let $n$ be a natural number. The largest prime power divisor of $n$, sometimes denoted $q(n)$ and sometimes denoted $a(n)$, is defined as the largest prime power that divides $n$.

Behavior

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

Upper bound

The value of ${\frac {q(n)}{n}}$ is largest when $n$ itself is a prime power, namely, it is $1$ for these values of $1$. Since there are infinitely many primes, we have:

$\lim \sup _{n\to \infty }{\frac {q(n)}{n}}=1$.

Lower bound

Further information: Largest prime power divisor has logarithmic lower bound

The largest prime power divisor of $n$ is $\Omega (\log n)$. In fact, we have:

$\lim \inf {\frac {q(n)}{\log n}}$ is finite and greater than zero.

Thus, we have:

$\lim _{n\to \infty }q(n)=\infty$.

Asymptotic fraction

Further information: Fractional distribution of largest prime power divisor

The value of $\log(q(n))/\log n$ is almost uniformly distributed in the interval $[0,1]$.

Relation with other arithmetic functions

• Prime divisor count function: This is the total number of prime divisors of $n$, and is denoted $\omega (n)$. We have the following relation:

$\omega (n)\log(q(n))\geq \log(n)$.