Properties of greatest integer function
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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
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
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up vote
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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
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
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
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
edited Jun 28 '15 at 23:48
iadvd
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
edited Jun 28 '15 at 23:48
iadvd
5,36692555
edited Jun 28 '15 at 23:48
iadvd
5,36692555
edited Jun 28 '15 at 23:48
iadvd
5,36692555
5,36692555
asked Jun 26 '15 at 21:30
Naveen Crasta
134
asked Jun 26 '15 at 21:30
Naveen Crasta
134
asked Jun 26 '15 at 21:30
Naveen Crasta
134
134
What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39
add a comment |Â
What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39
What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39
add a comment |Â
1 Answer
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- 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:
$$beginalign
&phi(n,k)=lfloor n rfloor-sum_p_ile kleftlfloor dfrac n (p_i)rightrfloor+sum_p_i<p_jle kleftlfloordfrac n(p_ip_j)rightrfloor-sum_p_i<p_j<p_mle kleftlfloor dfracn(p_ip_jp_m)rightrfloor+dots
endalign$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_k=1^inftylfloorfracnp^krfloor$$
- For positive integers $lfloor sqrtn+sqrtn+1rfloor=lfloorsqrt4n+2rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac-n2 rfloor=n$ 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 Vashishth
7,45411951
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
- 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:
$$beginalign
&phi(n,k)=lfloor n rfloor-sum_p_ile kleftlfloor dfrac n (p_i)rightrfloor+sum_p_i<p_jle kleftlfloordfrac n(p_ip_j)rightrfloor-sum_p_i<p_j<p_mle kleftlfloor dfracn(p_ip_jp_m)rightrfloor+dots
endalign$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_k=1^inftylfloorfracnp^krfloor$$
- For positive integers $lfloor sqrtn+sqrtn+1rfloor=lfloorsqrt4n+2rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac-n2 rfloor=n$ 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 Vashishth
7,45411951
add a comment |Â
up vote
0
down vote
- 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:
$$beginalign
&phi(n,k)=lfloor n rfloor-sum_p_ile kleftlfloor dfrac n (p_i)rightrfloor+sum_p_i<p_jle kleftlfloordfrac n(p_ip_j)rightrfloor-sum_p_i<p_j<p_mle kleftlfloor dfracn(p_ip_jp_m)rightrfloor+dots
endalign$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_k=1^inftylfloorfracnp^krfloor$$
- For positive integers $lfloor sqrtn+sqrtn+1rfloor=lfloorsqrt4n+2rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac-n2 rfloor=n$ 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 Vashishth
7,45411951
add a comment |Â
up vote
0
down vote
up vote
0
down vote
- 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:
$$beginalign
&phi(n,k)=lfloor n rfloor-sum_p_ile kleftlfloor dfrac n (p_i)rightrfloor+sum_p_i<p_jle kleftlfloordfrac n(p_ip_j)rightrfloor-sum_p_i<p_j<p_mle kleftlfloor dfracn(p_ip_jp_m)rightrfloor+dots
endalign$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_k=1^inftylfloorfracnp^krfloor$$
- For positive integers $lfloor sqrtn+sqrtn+1rfloor=lfloorsqrt4n+2rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac-n2 rfloor=n$ 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 Vashishth
7,45411951
- 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:
$$beginalign
&phi(n,k)=lfloor n rfloor-sum_p_ile kleftlfloor dfrac n (p_i)rightrfloor+sum_p_i<p_jle kleftlfloordfrac n(p_ip_j)rightrfloor-sum_p_i<p_j<p_mle kleftlfloor dfracn(p_ip_jp_m)rightrfloor+dots
endalign$$
which tells us that total number of times a prime $p$ divides $n!$ is $$sum_k=1^inftylfloorfracnp^krfloor$$
- For positive integers $lfloor sqrtn+sqrtn+1rfloor=lfloorsqrt4n+2rfloor$
- $lfloor2xrfloor+lfloor2yrfloorge lfloor x rfloor+lfloor y rfloor+lfloor x+y rfloor$
- $lfloor frac n2 rfloor- lfloor frac-n2 rfloor=n$ 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 Vashishth
7,45411951
answered Jun 29 '15 at 0:12
Bhaskar Vashishth
7,45411951
answered Jun 29 '15 at 0:12
Bhaskar Vashishth
7,45411951
answered Jun 29 '15 at 0:12
Bhaskar Vashishth
7,45411951
7,45411951
add a comment |Â
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What kind of properties are you looking for?
– AD.
Jun 28 '15 at 10:06
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 ?
– Naveen Crasta
Jun 28 '15 at 12:55
There is no such rule, as easy examples shows, please try to find some and you will see why.
– AD.
Jun 28 '15 at 17:39