[math-fun] Iterating the sum-of-divisors function
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x. My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k). Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
This paper at least has some interesting experimental facts about it: < http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ... >. ——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
a_2 = A007497, a_5 = A051572 will show you some initial terms. Please augment those entries with anything you find out! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This paper at least has some interesting experimental facts about it:
< http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
.
——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
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It seems that the sequence given in equation 4.2 of Cohen's paper that Dan referenced might belong in the OEIS (unless it already is and I missed it). As far as I can tell, it seems to be initial values of trajectories that are (up to their limit of experimentation) disjoint. The sequence begins: 2, 5, 16, 27, 29. 1 should probably be there as the trivial case. Kerry On Thu, Apr 30, 2015 at 5:23 PM, Neil Sloane <njasloane@gmail.com> wrote:
a_2 = A007497, a_5 = A051572 will show you some initial terms.
Please augment those entries with anything you find out!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This paper at least has some interesting experimental facts about it:
<
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
.
——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
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Kerry, Great - could you submit it? 1, 2, 5, 16, 27, 29, ... Thanks! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Apr 30, 2015 at 11:36 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
It seems that the sequence given in equation 4.2 of Cohen's paper that Dan referenced might belong in the OEIS (unless it already is and I missed it). As far as I can tell, it seems to be initial values of trajectories that are (up to their limit of experimentation) disjoint. The sequence begins: 2, 5, 16, 27, 29. 1 should probably be there as the trivial case.
Kerry
On Thu, Apr 30, 2015 at 5:23 PM, Neil Sloane <njasloane@gmail.com> wrote:
a_2 = A007497, a_5 = A051572 will show you some initial terms.
Please augment those entries with anything you find out!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This paper at least has some interesting experimental facts about it:
<
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
.
——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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That raises an interesting question. "Up to the limit of experimentation". There remains at least a philosophical possibility that some of these entries may merge when the exploration frontier is pushed forward. How would OEIS handle such an eventuality? One possibility is to define the sequence as initial values of trajectories that are disjoint up to some large but specified value, say 10^100. On Fri, May 1, 2015 at 1:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Kerry, Great - could you submit it? 1, 2, 5, 16, 27, 29, ...
Thanks!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 11:36 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
It seems that the sequence given in equation 4.2 of Cohen's paper that Dan referenced might belong in the OEIS (unless it already is and I missed it). As far as I can tell, it seems to be initial values of trajectories that are (up to their limit of experimentation) disjoint. The sequence begins: 2, 5, 16, 27, 29. 1 should probably be there as the trivial case.
Kerry
On Thu, Apr 30, 2015 at 5:23 PM, Neil Sloane <njasloane@gmail.com> wrote:
a_2 = A007497, a_5 = A051572 will show you some initial terms.
Please augment those entries with anything you find out!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This paper at least has some interesting experimental facts about it:
<
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
.
——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Since Allan points out that the sequence in question requires some delicate editorial decisions, I've created the entry myself - see A257348. Kerry, thanks for suggesting it. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, May 1, 2015 at 10:06 AM, Allan Wechsler <acwacw@gmail.com> wrote:
That raises an interesting question. "Up to the limit of experimentation". There remains at least a philosophical possibility that some of these entries may merge when the exploration frontier is pushed forward. How would OEIS handle such an eventuality? One possibility is to define the sequence as initial values of trajectories that are disjoint up to some large but specified value, say 10^100.
On Fri, May 1, 2015 at 1:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Kerry, Great - could you submit it? 1, 2, 5, 16, 27, 29, ...
Thanks!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 11:36 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
It seems that the sequence given in equation 4.2 of Cohen's paper that Dan referenced might belong in the OEIS (unless it already is and I missed it). As far as I can tell, it seems to be initial values of trajectories that are (up to their limit of experimentation) disjoint. The sequence begins: 2, 5, 16, 27, 29. 1 should probably be there as the trivial case.
Kerry
On Thu, Apr 30, 2015 at 5:23 PM, Neil Sloane <njasloane@gmail.com> wrote:
a_2 = A007497, a_5 = A051572 will show you some initial terms.
Please augment those entries with anything you find out!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This paper at least has some interesting experimental facts about it:
<
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
.
——Dan
On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x.
My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k).
Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
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Thanks, Neil. On Fri, May 1, 2015 at 8:25 AM, Neil Sloane <njasloane@gmail.com> wrote:
Since Allan points out that the sequence in question requires some delicate editorial decisions, I've created the entry myself - see A257348. Kerry, thanks for suggesting it.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Fri, May 1, 2015 at 10:06 AM, Allan Wechsler <acwacw@gmail.com> wrote:
That raises an interesting question. "Up to the limit of experimentation". There remains at least a philosophical possibility that some of these entries may merge when the exploration frontier is pushed forward. How would OEIS handle such an eventuality? One possibility is to define the sequence as initial values of trajectories that are disjoint up to some large but specified value, say 10^100.
On Fri, May 1, 2015 at 1:42 AM, Neil Sloane <njasloane@gmail.com> wrote:
Kerry, Great - could you submit it? 1, 2, 5, 16, 27, 29, ...
Thanks!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 11:36 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
It seems that the sequence given in equation 4.2 of Cohen's paper that Dan referenced might belong in the OEIS (unless it already is and I missed it). As far as I can tell, it seems to be initial values of trajectories that are (up to their limit of experimentation) disjoint. The sequence begins: 2, 5, 16, 27, 29. 1 should probably be there as the trivial case.
Kerry
On Thu, Apr 30, 2015 at 5:23 PM, Neil Sloane <njasloane@gmail.com> wrote:
a_2 = A007497, a_5 = A051572 will show you some initial terms.
Please augment those entries with anything you find out!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Apr 30, 2015 at 5:33 PM, Dan Asimov <dasimov@earthlink.net
wrote:
This paper at least has some interesting experimental facts about
it:
<
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.868&rep=rep1&typ...
>.
——Dan
> On Apr 30, 2015, at 1:59 PM, Allan Wechsler <acwacw@gmail.com> wrote: > > Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x. > > My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k). > > Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?
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participants (4)
-
Allan Wechsler -
Dan Asimov -
Kerry Mitchell -
Neil Sloane