Number of products of powers of primes less than n
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Is there a bound for the number of products of powers of a set of prime numbers less than a given number n? For instance, if I am given the set of prime numbers {2,7,11,13} and I am only interested in products of powers of primes less than 32, then there are twelve products of powers of these primes numbers: $2=2^1$, $4=2^2$, $7=7^1$, $8=2^3$, $11=11^1$, $13=13^1$, $14=2^1cdot 7^1$, $16=2^4$, $22=2^1cdot 11^1$, $26=2^1cdot 13^1$, $28=2^2cdot 7^1$, and $32=2^5$.
I know people have asked about estimating the number of products of primes less than n and estimating the number of prime powers less than n. I am looking for a combination of these two questions.
number-theory elementary-number-theory
asked Dec 5 at 12:11
John Asplund
386
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up vote
1
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favorite
Is there a bound for the number of products of powers of a set of prime numbers less than a given number n? For instance, if I am given the set of prime numbers {2,7,11,13} and I am only interested in products of powers of primes less than 32, then there are twelve products of powers of these primes numbers: $2=2^1$, $4=2^2$, $7=7^1$, $8=2^3$, $11=11^1$, $13=13^1$, $14=2^1cdot 7^1$, $16=2^4$, $22=2^1cdot 11^1$, $26=2^1cdot 13^1$, $28=2^2cdot 7^1$, and $32=2^5$.
I know people have asked about estimating the number of products of primes less than n and estimating the number of prime powers less than n. I am looking for a combination of these two questions.
number-theory elementary-number-theory
asked Dec 5 at 12:11
John Asplund
386
N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is there a bound for the number of products of powers of a set of prime numbers less than a given number n? For instance, if I am given the set of prime numbers {2,7,11,13} and I am only interested in products of powers of primes less than 32, then there are twelve products of powers of these primes numbers: $2=2^1$, $4=2^2$, $7=7^1$, $8=2^3$, $11=11^1$, $13=13^1$, $14=2^1cdot 7^1$, $16=2^4$, $22=2^1cdot 11^1$, $26=2^1cdot 13^1$, $28=2^2cdot 7^1$, and $32=2^5$.
I know people have asked about estimating the number of products of primes less than n and estimating the number of prime powers less than n. I am looking for a combination of these two questions.
number-theory elementary-number-theory
asked Dec 5 at 12:11
John Asplund
386
Is there a bound for the number of products of powers of a set of prime numbers less than a given number n? For instance, if I am given the set of prime numbers {2,7,11,13} and I am only interested in products of powers of primes less than 32, then there are twelve products of powers of these primes numbers: $2=2^1$, $4=2^2$, $7=7^1$, $8=2^3$, $11=11^1$, $13=13^1$, $14=2^1cdot 7^1$, $16=2^4$, $22=2^1cdot 11^1$, $26=2^1cdot 13^1$, $28=2^2cdot 7^1$, and $32=2^5$.
I know people have asked about estimating the number of products of primes less than n and estimating the number of prime powers less than n. I am looking for a combination of these two questions.
number-theory elementary-number-theory
number-theory elementary-number-theory
asked Dec 5 at 12:11
John Asplund
386
asked Dec 5 at 12:11
John Asplund
386
edited Dec 5 at 17:35
asked Dec 5 at 12:11
John Asplund
386
asked Dec 5 at 12:11
John Asplund
386
asked Dec 5 at 12:11
John Asplund
386
386
N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31
add a comment |
N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31
N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31
add a comment |
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I thougth of this : you want to maximize the function Z = 2$^a$ + 7$^b$ + 11$^c$ + 13$^d$ , whenever z $leq$ n. You may "linearize" this function by taking natural logarithm.
By means of linear simplex method, you may find the máxima...
Then you may try and analize which 4 - tuples (a,b,c,d) with integer coordinates are nearest the máxima found . Since exponentiating and taking logarithm are continous, taking "nearby" tuples should work.
I will try to find a more combinatorial answer, though. It is a very interesting question.
answered Dec 5 at 18:14
vanmeri
658
add a comment |
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I thougth of this : you want to maximize the function Z = 2$^a$ + 7$^b$ + 11$^c$ + 13$^d$ , whenever z $leq$ n. You may "linearize" this function by taking natural logarithm.
By means of linear simplex method, you may find the máxima...
Then you may try and analize which 4 - tuples (a,b,c,d) with integer coordinates are nearest the máxima found . Since exponentiating and taking logarithm are continous, taking "nearby" tuples should work.
I will try to find a more combinatorial answer, though. It is a very interesting question.
answered Dec 5 at 18:14
vanmeri
658
add a comment |
up vote
1
down vote
I thougth of this : you want to maximize the function Z = 2$^a$ + 7$^b$ + 11$^c$ + 13$^d$ , whenever z $leq$ n. You may "linearize" this function by taking natural logarithm.
By means of linear simplex method, you may find the máxima...
Then you may try and analize which 4 - tuples (a,b,c,d) with integer coordinates are nearest the máxima found . Since exponentiating and taking logarithm are continous, taking "nearby" tuples should work.
I will try to find a more combinatorial answer, though. It is a very interesting question.
answered Dec 5 at 18:14
vanmeri
658
add a comment |
up vote
1
down vote
up vote
1
down vote
I thougth of this : you want to maximize the function Z = 2$^a$ + 7$^b$ + 11$^c$ + 13$^d$ , whenever z $leq$ n. You may "linearize" this function by taking natural logarithm.
By means of linear simplex method, you may find the máxima...
Then you may try and analize which 4 - tuples (a,b,c,d) with integer coordinates are nearest the máxima found . Since exponentiating and taking logarithm are continous, taking "nearby" tuples should work.
I will try to find a more combinatorial answer, though. It is a very interesting question.
answered Dec 5 at 18:14
vanmeri
658
I thougth of this : you want to maximize the function Z = 2$^a$ + 7$^b$ + 11$^c$ + 13$^d$ , whenever z $leq$ n. You may "linearize" this function by taking natural logarithm.
By means of linear simplex method, you may find the máxima...
Then you may try and analize which 4 - tuples (a,b,c,d) with integer coordinates are nearest the máxima found . Since exponentiating and taking logarithm are continous, taking "nearby" tuples should work.
I will try to find a more combinatorial answer, though. It is a very interesting question.
answered Dec 5 at 18:14
vanmeri
658
answered Dec 5 at 18:14
vanmeri
658
answered Dec 5 at 18:14
vanmeri
658
answered Dec 5 at 18:14
vanmeri
658
658
add a comment |
add a comment |
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N is given, are the primes given as well?Or you mean ANY set of primes under n?
– vanmeri
Dec 5 at 13:07
If the primes aren't given, then you would end up knowing quantity of primes under every natural!
– vanmeri
Dec 5 at 13:09
I guess I was not clear enough. If I am given a set of prime numbers as I did above {2,7,11,13}, how many numbers are a product of powers of those prime numbers? In the example above, there are twelve. I will edit the problem to reflect my original intention.
– John Asplund
Dec 5 at 17:31