Properties of greatest integer function
0
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I am curious to know some properties of the floor functions, for instance, $lfloor a cdot x rfloor$, $lfloor a1cdot x1+a2cdot x2 rfloor$, etc. Is there any book that contains such properties ?
special-functions
edited Jun 28 '15 at 23:48
iadvd
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
$endgroup$
add a comment |
0
$begingroup$
I am curious to know some properties of the floor functions, for instance, $lfloor a cdot x rfloor$, $lfloor a1cdot x1+a2cdot x2 rfloor$, etc. Is there any book that contains such properties ?
special-functions
edited Jun 28 '15 at 23:48
iadvd
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
$endgroup$
$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39
add a comment |
0
0
0
$begingroup$
I am curious to know some properties of the floor functions, for instance, $lfloor a cdot x rfloor$, $lfloor a1cdot x1+a2cdot x2 rfloor$, etc. Is there any book that contains such properties ?
special-functions
edited Jun 28 '15 at 23:48
iadvd
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
$endgroup$
I am curious to know some properties of the floor functions, for instance, $lfloor a cdot x rfloor$, $lfloor a1cdot x1+a2cdot x2 rfloor$, etc. Is there any book that contains such properties ?
special-functions
special-functions
edited Jun 28 '15 at 23:48
iadvd
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
edited Jun 28 '15 at 23:48
iadvd
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
edited Jun 28 '15 at 23:48
iadvd
5,452102656
edited Jun 28 '15 at 23:48
iadvd
5,452102656
edited Jun 28 '15 at 23:48
iadvd
5,452102656
5,452102656
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
asked Jun 26 '15 at 21:30
Naveen CrastaNaveen Crasta
134
134
$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39
add a comment |
$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39
$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39
add a comment |
1 Answer
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$begingroup$
- There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes:
$$begin{align}
&phi(n,k)=lfloor n rfloor-sum_{p_ile k}leftlfloor dfrac{ n }{(p_i)}rightrfloor+sum_{p_i<p_jle k}leftlfloordfrac{ n}{(p_ip_j)}rightrfloor-sum_{p_i<p_j<p_mle k}leftlfloor dfrac{n}{(p_ip_jp_m)}rightrfloor+dots
end{align}$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_{k=1}^{infty}lfloorfrac{n}{p^k}rfloor$$
- For positive integers $lfloor sqrt{n}+sqrt{n+1}rfloor=lfloorsqrt{4n+2}rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac{-n}{2} rfloor=n$ for integers $n ge 0$.
You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
$endgroup$
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
add a comment |
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1 Answer
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1 Answer
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0
$begingroup$
- There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes:
$$begin{align}
&phi(n,k)=lfloor n rfloor-sum_{p_ile k}leftlfloor dfrac{ n }{(p_i)}rightrfloor+sum_{p_i<p_jle k}leftlfloordfrac{ n}{(p_ip_j)}rightrfloor-sum_{p_i<p_j<p_mle k}leftlfloor dfrac{n}{(p_ip_jp_m)}rightrfloor+dots
end{align}$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_{k=1}^{infty}lfloorfrac{n}{p^k}rfloor$$
- For positive integers $lfloor sqrt{n}+sqrt{n+1}rfloor=lfloorsqrt{4n+2}rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac{-n}{2} rfloor=n$ for integers $n ge 0$.
You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
$endgroup$
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
add a comment |
0
$begingroup$
- There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes:
$$begin{align}
&phi(n,k)=lfloor n rfloor-sum_{p_ile k}leftlfloor dfrac{ n }{(p_i)}rightrfloor+sum_{p_i<p_jle k}leftlfloordfrac{ n}{(p_ip_j)}rightrfloor-sum_{p_i<p_j<p_mle k}leftlfloor dfrac{n}{(p_ip_jp_m)}rightrfloor+dots
end{align}$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_{k=1}^{infty}lfloorfrac{n}{p^k}rfloor$$
- For positive integers $lfloor sqrt{n}+sqrt{n+1}rfloor=lfloorsqrt{4n+2}rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac{-n}{2} rfloor=n$ for integers $n ge 0$.
You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
$endgroup$
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
add a comment |
0
0
0
$begingroup$
- There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes:
$$begin{align}
&phi(n,k)=lfloor n rfloor-sum_{p_ile k}leftlfloor dfrac{ n }{(p_i)}rightrfloor+sum_{p_i<p_jle k}leftlfloordfrac{ n}{(p_ip_j)}rightrfloor-sum_{p_i<p_j<p_mle k}leftlfloor dfrac{n}{(p_ip_jp_m)}rightrfloor+dots
end{align}$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_{k=1}^{infty}lfloorfrac{n}{p^k}rfloor$$
- For positive integers $lfloor sqrt{n}+sqrt{n+1}rfloor=lfloorsqrt{4n+2}rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac{-n}{2} rfloor=n$ for integers $n ge 0$.
You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
$endgroup$
- There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes:
$$begin{align}
&phi(n,k)=lfloor n rfloor-sum_{p_ile k}leftlfloor dfrac{ n }{(p_i)}rightrfloor+sum_{p_i<p_jle k}leftlfloordfrac{ n}{(p_ip_j)}rightrfloor-sum_{p_i<p_j<p_mle k}leftlfloor dfrac{n}{(p_ip_jp_m)}rightrfloor+dots
end{align}$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_{k=1}^{infty}lfloorfrac{n}{p^k}rfloor$$
- For positive integers $lfloor sqrt{n}+sqrt{n+1}rfloor=lfloorsqrt{4n+2}rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac{-n}{2} rfloor=n$ for integers $n ge 0$.
You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
answered Jun 29 '15 at 0:12
Bhaskar VashishthBhaskar Vashishth
7,67212054
7,67212054
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
add a comment |
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
$begingroup$
Also, you can refer the following : math.slu.edu/~clair/mcmc/Floor-solutions.pdf
$endgroup$
– SARTHAK GUPTA
Dec 23 '18 at 11:00
add a comment |
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$begingroup$
What kind of properties are you looking for?
$endgroup$
– AD.
Jun 28 '15 at 10:06
$begingroup$
For example, is it true that floor(ax) = afloor(x) for non-negative real number a ? What can you tell if a is negative real ?
$endgroup$
– Naveen Crasta
Jun 28 '15 at 12:55
$begingroup$
There is no such rule, as easy examples shows, please try to find some and you will see why.
$endgroup$
– AD.
Jun 28 '15 at 17:39