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 greater than <math>\log n</math>. Thus, we have:
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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

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

.