Selfish Primes Question
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Consider the series : $1,2,4,8,16,23,28,38,49,ldots $
( number + sum of its digits numbers gives the next number and continues up to infinity )
Now, consider the primes in this series ( the first ones $2,23,101,103,107 , ldots $)
Now, can you help with the following questions :
1) what is the reciprocal sum of primes in this series? ( my guess is around 0.7 which makes the series converges but how I can prove that ? )
2) How many consecutive primes can be found in this series? ( My guess is 7 or 8 , but still I am not able to prove this result, for example, 101,103,107 are the consecutive primes in the series )
3) It is believed that $fracx(ln x)^2$ roughly gives the number of twin primes and I suspect that that formula works for the number of primes in the series ( $1,2,4,8,16,23,28,38,ldots$ )
Any help will be appreciated. Thank you.
number-theory
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
asked Sep 9 at 11:06
Turker
6
add a comment |Â
up vote
0
down vote
favorite
Consider the series : $1,2,4,8,16,23,28,38,49,ldots $
( number + sum of its digits numbers gives the next number and continues up to infinity )
Now, consider the primes in this series ( the first ones $2,23,101,103,107 , ldots $)
Now, can you help with the following questions :
1) what is the reciprocal sum of primes in this series? ( my guess is around 0.7 which makes the series converges but how I can prove that ? )
2) How many consecutive primes can be found in this series? ( My guess is 7 or 8 , but still I am not able to prove this result, for example, 101,103,107 are the consecutive primes in the series )
3) It is believed that $fracx(ln x)^2$ roughly gives the number of twin primes and I suspect that that formula works for the number of primes in the series ( $1,2,4,8,16,23,28,38,ldots$ )
Any help will be appreciated. Thank you.
number-theory
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
asked Sep 9 at 11:06
Turker
6
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
1
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
2
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the series : $1,2,4,8,16,23,28,38,49,ldots $
( number + sum of its digits numbers gives the next number and continues up to infinity )
Now, consider the primes in this series ( the first ones $2,23,101,103,107 , ldots $)
Now, can you help with the following questions :
1) what is the reciprocal sum of primes in this series? ( my guess is around 0.7 which makes the series converges but how I can prove that ? )
2) How many consecutive primes can be found in this series? ( My guess is 7 or 8 , but still I am not able to prove this result, for example, 101,103,107 are the consecutive primes in the series )
3) It is believed that $fracx(ln x)^2$ roughly gives the number of twin primes and I suspect that that formula works for the number of primes in the series ( $1,2,4,8,16,23,28,38,ldots$ )
Any help will be appreciated. Thank you.
number-theory
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
asked Sep 9 at 11:06
Turker
6
Consider the series : $1,2,4,8,16,23,28,38,49,ldots $
( number + sum of its digits numbers gives the next number and continues up to infinity )
Now, consider the primes in this series ( the first ones $2,23,101,103,107 , ldots $)
Now, can you help with the following questions :
1) what is the reciprocal sum of primes in this series? ( my guess is around 0.7 which makes the series converges but how I can prove that ? )
2) How many consecutive primes can be found in this series? ( My guess is 7 or 8 , but still I am not able to prove this result, for example, 101,103,107 are the consecutive primes in the series )
3) It is believed that $fracx(ln x)^2$ roughly gives the number of twin primes and I suspect that that formula works for the number of primes in the series ( $1,2,4,8,16,23,28,38,ldots$ )
Any help will be appreciated. Thank you.
number-theory
number-theory
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
asked Sep 9 at 11:06
Turker
6
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
asked Sep 9 at 11:06
Turker
6
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
edited Sep 9 at 11:25
José Carlos Santos
123k17101186
123k17101186
asked Sep 9 at 11:06
Turker
6
asked Sep 9 at 11:06
Turker
6
asked Sep 9 at 11:06
Turker
6
6
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
1
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
2
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11
add a comment |Â
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
1
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
2
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
1
1
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
2
2
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2910679%2fselfish-primes-question%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Sep 9 at 11:11
1
Where did this question come from? Are you taking a course? Doing research? Or is this just for fun?
– Theo Bendit
Sep 9 at 11:19
I have a master degree in number theory. The series actually known as Kaprekar series ( Indian math high school teacher lived 1950's) . I have been searching the answers of my question almost 2 years.
– Turker
Sep 9 at 11:28
2
3) seems plausible, since the sum of digits of $x$ is roughly $frac5ln 10ln x$, so the density of the sequence at $x$ is proportional to $frac1ln x$, so the density of primes in the sequence is proportional to $frac1(ln x)^2$. That would also imply that the sum of their reciprocals converges. I don't think you have a snowball's chance in hell to find the limit in closed form; I'm surprised you're hoping for that after doing a master in number theory.
– joriki
Sep 9 at 11:40
I just posted only 3 questions but more interesting questions related to this series about to come.
– Turker
Sep 9 at 18:11