Largest prime power divisor - Revision history

Vipul at 02:08, 29 April 2009

2009-04-29T02:08:38Z

← Older revision		Revision as of 02:08, 29 April 2009
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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:		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:

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

	===Lower bound===		===Lower bound===
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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>.

			==Relation with other arithmetic functions==

			* [[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:

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

Vipul at 00:00, 29 April 2009

2009-04-29T00:00:39Z

← Older revision		Revision as of 00:00, 29 April 2009
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	{{oeis\|A034699}}		{{oeis\|A034699}}

			===Upper bound===

			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:

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

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

			Thus, we have:

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

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $n$ be a natural number. The '''largest prime power divisor''' of $n$ , sometimes denoted $q(n)$ and ...'

2009-04-28T22:51:52Z

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 ...'

New page

{{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 sometimes denoted <math>a(n)</math>, is defined as the largest prime power that divides <math>n</math>.

==Behavior==

{{oeis|A034699}}

===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:

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

===Asymptotic fraction===

{{further|[[Fractional distribution of largest prime power divisor]]}}

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

Largest prime power divisor - Revision history

Vipul at 02:08, 29 April 2009

Vipul at 00:00, 29 April 2009

Vipul: Created page with '{{arithmetic function}} ==Definition== Let n be a natural number. The '''largest prime power divisor''' of n, sometimes denoted q(n) and ...'

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $n$ be a natural number. The '''largest prime power divisor''' of $n$ , sometimes denoted $q(n)$ and ...'