# Difference between revisions of "Largest prime power divisor"

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(Created page with '{{arithmetic function}} ==Definition== Let <math>n</math> be a natural number. The '''largest prime power divisor''' of <math>n</math>, sometimes denoted <math>q(n)</math> and ...') |
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{{oeis|A034699}} | {{oeis|A034699}} | ||

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+ | ===Upper bound=== | ||

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+ | The value of <math>\frac{q(n)}{n}</math> is largest when <math>n</math> itself is a prime power, namely, it is <math>1</math> for these values of <math>1</math>. Since there are [[infinitude of primes|infinitely many primes]], we have: | ||

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+ | <math>\lim \sup_{n \to \infty} \frac{q(n)}{n} = 1</math>. | ||

===Lower bound=== | ===Lower bound=== | ||

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{{further|[[Largest prime power divisor has logarithmic lower bound]]}} | {{further|[[Largest prime power divisor has logarithmic lower bound]]}} | ||

− | The largest prime power divisor of <math>n</math> is | + | The largest prime power divisor of <math>n</math> is <math>\Omega(\log n)</math>. In fact, we have: |

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+ | <math>\lim \inf \frac{q(n)}{\log n}</math> is finite and greater than zero. | ||

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+ | Thus, we have: | ||

<math>\lim_{n \to \infty} q(n) = \infty</math>. | <math>\lim_{n \to \infty} q(n) = \infty</math>. | ||

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

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+ | ==Relation with other arithmetic functions== | ||

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+ | * [[Prime divisor count function]]: This is the total number of prime divisors of <math>n</math>, and is denoted <math>\omega(n)</math>. We have the following relation: | ||

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+ | <math>\omega(n)\log(q(n)) \ge \log(n)</math>. |

## Latest revision as of 02:08, 29 April 2009

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:

.