Mixing
Gaps and density of numbers consisting of sums of products of primes where the number of factors for each...
$begingroup$
For the following set of numbers:
$$ { n } =sum_{i=1}^{infty} b_i p_i^{p_{j_i}} $$
where each $b_i$ (b for binary) is either 1 or zero
each $n$ in the set ${ n }$ has a unique set of ${b_i}$ and ${j_i}$
Is there an anticipated asymptotic density for all possible $n$ thus defined?
first several lowest such $n$ start as 4, 8, 9, 13, 17, 25, 27, 29, 31, 32, 33...
this is $2^2,2^3,3^2,3^2+2^2,3^2+2^3,5^2, 3^3, 5^2+2^2, 3^3+2^2, 2^5, 5^2+2^3$, etc
sequences-and-series prime-numbers pseudoprimes
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
$endgroup$
add a comment |
$begingroup$
For the following set of numbers:
$$ { n } =sum_{i=1}^{infty} b_i p_i^{p_{j_i}} $$
where each $b_i$ (b for binary) is either 1 or zero
each $n$ in the set ${ n }$ has a unique set of ${b_i}$ and ${j_i}$
Is there an anticipated asymptotic density for all possible $n$ thus defined?
first several lowest such $n$ start as 4, 8, 9, 13, 17, 25, 27, 29, 31, 32, 33...
this is $2^2,2^3,3^2,3^2+2^2,3^2+2^3,5^2, 3^3, 5^2+2^2, 3^3+2^2, 2^5, 5^2+2^3$, etc
sequences-and-series prime-numbers pseudoprimes
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
$endgroup$
1
$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09
add a comment |
$begingroup$
For the following set of numbers:
$$ { n } =sum_{i=1}^{infty} b_i p_i^{p_{j_i}} $$
where each $b_i$ (b for binary) is either 1 or zero
each $n$ in the set ${ n }$ has a unique set of ${b_i}$ and ${j_i}$
Is there an anticipated asymptotic density for all possible $n$ thus defined?
first several lowest such $n$ start as 4, 8, 9, 13, 17, 25, 27, 29, 31, 32, 33...
this is $2^2,2^3,3^2,3^2+2^2,3^2+2^3,5^2, 3^3, 5^2+2^2, 3^3+2^2, 2^5, 5^2+2^3$, etc
sequences-and-series prime-numbers pseudoprimes
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
$endgroup$
For the following set of numbers:
$$ { n } =sum_{i=1}^{infty} b_i p_i^{p_{j_i}} $$
where each $b_i$ (b for binary) is either 1 or zero
each $n$ in the set ${ n }$ has a unique set of ${b_i}$ and ${j_i}$
Is there an anticipated asymptotic density for all possible $n$ thus defined?
first several lowest such $n$ start as 4, 8, 9, 13, 17, 25, 27, 29, 31, 32, 33...
this is $2^2,2^3,3^2,3^2+2^2,3^2+2^3,5^2, 3^3, 5^2+2^2, 3^3+2^2, 2^5, 5^2+2^3$, etc
sequences-and-series prime-numbers pseudoprimes
sequences-and-series prime-numbers pseudoprimes
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
edited Jan 23 at 18:39
phdmba7of12
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
asked Jan 21 at 14:59
phdmba7of12phdmba7of12
223419
223419
1
$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09
add a comment |
1
$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09
1
1
$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09
add a comment |
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$begingroup$
I don't see what you mean with "has a unique set of $b_i,j_i$". Try replacing $p_k $ by $lceil k log krceil$ see what you get and if it is close. If not then probably you can't say anything about the density. Also replacing the exponent by $2$ shouldn't change your question (and its result) a lot.
$endgroup$
– reuns
Jan 21 at 20:33
$begingroup$
to clarify, each item in the set is a sum of powers of primes, where each exponent itself a prime ... look at the example above and see if this makes sense and/or if there is a better way to notate the concept.
$endgroup$
– phdmba7of12
Jan 21 at 22:03
$begingroup$
as a test if it is clear, try to list the next three lowest numbers in the set
$endgroup$
– phdmba7of12
Jan 21 at 22:09