Iterated arithmetic derivative
$begingroup$
I found by accident the notion of arithmetic derivative. It goes as follows: if $p$ is prime then $p' = 1$ and it follows the usual Leibniz rule
$$
(p q)' = p' q + p q'quadforall ;p,q inmathbb{N},.
$$
My question regards the iterated action of this derivative on natural numbers, that is, I want to study the behaviour of
$$
n^{(k)} equiv n^{overset{k}{overbrace{primeldotsprime}}},.
$$
This was inspired by a question in this video (which is where I found out about this).
I am not a matematician so I don't have many tools to study this. So far I just did a table with Mathematica. It seems that powers of 2 tend to diverge to infinity (apart from $4$ which is a fixed point and $2$ which is prime). And I also observed that some numbers tend to infinity sharing the same trajectory, for example
$$
160^{(5)} = 180^{(5)} = 4834,,
$$
and diverge together from then onward. Another simple observation is the for any prime $p$, $(p^p)' = p^p$. So there are infinitely many fixed points.
Is there anything about this in the literature? Are there fixed points other than $p^p$? What is the rate of growth of $n^{(k)}$ when it doesn't reach zero? Is there a criterion to know in advance if a number is going to reach zero? Even a partial answer to any of these questions will be appreciated. I attach a log-log plot of the result of 30 iterations of the first $65536$ numbers. Red dots indicate powers of $2$.
Log-Log plot of the first 65536 numbers
number-theory arithmetic-derivative
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
$endgroup$
add a comment |
$begingroup$
I found by accident the notion of arithmetic derivative. It goes as follows: if $p$ is prime then $p' = 1$ and it follows the usual Leibniz rule
$$
(p q)' = p' q + p q'quadforall ;p,q inmathbb{N},.
$$
My question regards the iterated action of this derivative on natural numbers, that is, I want to study the behaviour of
$$
n^{(k)} equiv n^{overset{k}{overbrace{primeldotsprime}}},.
$$
This was inspired by a question in this video (which is where I found out about this).
I am not a matematician so I don't have many tools to study this. So far I just did a table with Mathematica. It seems that powers of 2 tend to diverge to infinity (apart from $4$ which is a fixed point and $2$ which is prime). And I also observed that some numbers tend to infinity sharing the same trajectory, for example
$$
160^{(5)} = 180^{(5)} = 4834,,
$$
and diverge together from then onward. Another simple observation is the for any prime $p$, $(p^p)' = p^p$. So there are infinitely many fixed points.
Is there anything about this in the literature? Are there fixed points other than $p^p$? What is the rate of growth of $n^{(k)}$ when it doesn't reach zero? Is there a criterion to know in advance if a number is going to reach zero? Even a partial answer to any of these questions will be appreciated. I attach a log-log plot of the result of 30 iterations of the first $65536$ numbers. Red dots indicate powers of $2$.
Log-Log plot of the first 65536 numbers
number-theory arithmetic-derivative
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
$endgroup$
1
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
1
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05
add a comment |
$begingroup$
I found by accident the notion of arithmetic derivative. It goes as follows: if $p$ is prime then $p' = 1$ and it follows the usual Leibniz rule
$$
(p q)' = p' q + p q'quadforall ;p,q inmathbb{N},.
$$
My question regards the iterated action of this derivative on natural numbers, that is, I want to study the behaviour of
$$
n^{(k)} equiv n^{overset{k}{overbrace{primeldotsprime}}},.
$$
This was inspired by a question in this video (which is where I found out about this).
I am not a matematician so I don't have many tools to study this. So far I just did a table with Mathematica. It seems that powers of 2 tend to diverge to infinity (apart from $4$ which is a fixed point and $2$ which is prime). And I also observed that some numbers tend to infinity sharing the same trajectory, for example
$$
160^{(5)} = 180^{(5)} = 4834,,
$$
and diverge together from then onward. Another simple observation is the for any prime $p$, $(p^p)' = p^p$. So there are infinitely many fixed points.
Is there anything about this in the literature? Are there fixed points other than $p^p$? What is the rate of growth of $n^{(k)}$ when it doesn't reach zero? Is there a criterion to know in advance if a number is going to reach zero? Even a partial answer to any of these questions will be appreciated. I attach a log-log plot of the result of 30 iterations of the first $65536$ numbers. Red dots indicate powers of $2$.
Log-Log plot of the first 65536 numbers
number-theory arithmetic-derivative
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
$endgroup$
I found by accident the notion of arithmetic derivative. It goes as follows: if $p$ is prime then $p' = 1$ and it follows the usual Leibniz rule
$$
(p q)' = p' q + p q'quadforall ;p,q inmathbb{N},.
$$
My question regards the iterated action of this derivative on natural numbers, that is, I want to study the behaviour of
$$
n^{(k)} equiv n^{overset{k}{overbrace{primeldotsprime}}},.
$$
This was inspired by a question in this video (which is where I found out about this).
I am not a matematician so I don't have many tools to study this. So far I just did a table with Mathematica. It seems that powers of 2 tend to diverge to infinity (apart from $4$ which is a fixed point and $2$ which is prime). And I also observed that some numbers tend to infinity sharing the same trajectory, for example
$$
160^{(5)} = 180^{(5)} = 4834,,
$$
and diverge together from then onward. Another simple observation is the for any prime $p$, $(p^p)' = p^p$. So there are infinitely many fixed points.
Is there anything about this in the literature? Are there fixed points other than $p^p$? What is the rate of growth of $n^{(k)}$ when it doesn't reach zero? Is there a criterion to know in advance if a number is going to reach zero? Even a partial answer to any of these questions will be appreciated. I attach a log-log plot of the result of 30 iterations of the first $65536$ numbers. Red dots indicate powers of $2$.
Log-Log plot of the first 65536 numbers
number-theory arithmetic-derivative
number-theory arithmetic-derivative
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
edited Dec 16 '18 at 8:07
Mane.andrea
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
asked Dec 15 '18 at 21:48
Mane.andreaMane.andrea
1265
1265
1
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
1
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05
add a comment |
1
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
1
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05
1
1
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
1
1
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05
add a comment |
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1
$begingroup$
Relevant: en.m.wikipedia.org/wiki/General_Leibniz_rule
$endgroup$
– JavaMan
Dec 15 '18 at 21:58
1
$begingroup$
Let $f(n) = sum_{p^k | n} frac{k}{p}$. Then $f(nm) = f(n)+f(m)$ is completely additive and you are looking at $D(n) = f(n) n$. You can extend to rationals with $f(n/m) = f(n)-f(m)$ and $D(x) = f(x) x$. For any completely additive function $g:mathbb{Q} to mathbb{Q}$ then you can look at $x mapsto g(x) x$. Those things are not linear so @JavaMan's comment doesn't apply.
$endgroup$
– reuns
Dec 16 '18 at 0:05