Maximum of least common multiple of numbers whose sum is fixed
$begingroup$
Let $n in mathbb{N}$ be given. What is the maximal value of the least common multiple of $x_1, ldots, x_k$, all positive integers, when
$$x_1 + ldots + x_k = n : ,$$
and what is the asymptotic value of $max( mathrm{LCM}( x_1, ldots, x_k))$?
Here's what I've found: We need to set $x_1, ldots , x_k$ to as many prime numbers as possible. So if, say, $n=5$, we set $5 = 2 + 3 $ with $mathrm{LCM}(2,3) =6$, and if $n=10$, we set $10 = 2+3+5$, $mathrm{LCM}(2,3,5)=30$.
The sum of all prime numbers less than $x$ are
$$sum_{p leq x} p approx mathrm{Li} (x^2) ,$$
(see What is the sum of the prime numbers up to a prime number $n$? , where Li is the logarithmic integral), and the product of prime numbers less than $x$ is given by the primorial function:
$$prod_{p leq x } p approx exp{x}$$
(see https://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers ). Combining these formulas, I get
$$max( mathrm{LCM}( x_1, ldots, x_k) approx exp left( sqrt{ mathrm{Li}^{-1} ( n) } right) $$
but is this correct?
elementary-number-theory prime-numbers
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
$endgroup$
add a comment |
$begingroup$
Let $n in mathbb{N}$ be given. What is the maximal value of the least common multiple of $x_1, ldots, x_k$, all positive integers, when
$$x_1 + ldots + x_k = n : ,$$
and what is the asymptotic value of $max( mathrm{LCM}( x_1, ldots, x_k))$?
Here's what I've found: We need to set $x_1, ldots , x_k$ to as many prime numbers as possible. So if, say, $n=5$, we set $5 = 2 + 3 $ with $mathrm{LCM}(2,3) =6$, and if $n=10$, we set $10 = 2+3+5$, $mathrm{LCM}(2,3,5)=30$.
The sum of all prime numbers less than $x$ are
$$sum_{p leq x} p approx mathrm{Li} (x^2) ,$$
(see What is the sum of the prime numbers up to a prime number $n$? , where Li is the logarithmic integral), and the product of prime numbers less than $x$ is given by the primorial function:
$$prod_{p leq x } p approx exp{x}$$
(see https://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers ). Combining these formulas, I get
$$max( mathrm{LCM}( x_1, ldots, x_k) approx exp left( sqrt{ mathrm{Li}^{-1} ( n) } right) $$
but is this correct?
elementary-number-theory prime-numbers
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
$endgroup$
add a comment |
$begingroup$
Let $n in mathbb{N}$ be given. What is the maximal value of the least common multiple of $x_1, ldots, x_k$, all positive integers, when
$$x_1 + ldots + x_k = n : ,$$
and what is the asymptotic value of $max( mathrm{LCM}( x_1, ldots, x_k))$?
Here's what I've found: We need to set $x_1, ldots , x_k$ to as many prime numbers as possible. So if, say, $n=5$, we set $5 = 2 + 3 $ with $mathrm{LCM}(2,3) =6$, and if $n=10$, we set $10 = 2+3+5$, $mathrm{LCM}(2,3,5)=30$.
The sum of all prime numbers less than $x$ are
$$sum_{p leq x} p approx mathrm{Li} (x^2) ,$$
(see What is the sum of the prime numbers up to a prime number $n$? , where Li is the logarithmic integral), and the product of prime numbers less than $x$ is given by the primorial function:
$$prod_{p leq x } p approx exp{x}$$
(see https://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers ). Combining these formulas, I get
$$max( mathrm{LCM}( x_1, ldots, x_k) approx exp left( sqrt{ mathrm{Li}^{-1} ( n) } right) $$
but is this correct?
elementary-number-theory prime-numbers
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
$endgroup$
Let $n in mathbb{N}$ be given. What is the maximal value of the least common multiple of $x_1, ldots, x_k$, all positive integers, when
$$x_1 + ldots + x_k = n : ,$$
and what is the asymptotic value of $max( mathrm{LCM}( x_1, ldots, x_k))$?
Here's what I've found: We need to set $x_1, ldots , x_k$ to as many prime numbers as possible. So if, say, $n=5$, we set $5 = 2 + 3 $ with $mathrm{LCM}(2,3) =6$, and if $n=10$, we set $10 = 2+3+5$, $mathrm{LCM}(2,3,5)=30$.
The sum of all prime numbers less than $x$ are
$$sum_{p leq x} p approx mathrm{Li} (x^2) ,$$
(see What is the sum of the prime numbers up to a prime number $n$? , where Li is the logarithmic integral), and the product of prime numbers less than $x$ is given by the primorial function:
$$prod_{p leq x } p approx exp{x}$$
(see https://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers ). Combining these formulas, I get
$$max( mathrm{LCM}( x_1, ldots, x_k) approx exp left( sqrt{ mathrm{Li}^{-1} ( n) } right) $$
but is this correct?
elementary-number-theory prime-numbers
elementary-number-theory prime-numbers
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
asked Jan 20 at 19:00
JAskgaardJAskgaard
1367
1367
add a comment |
add a comment |
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