[math-fun] Transforming integer D into integer E (exponentiation)
Hello Math-Fun, Say D has 5 digits: D = abcde Starting from the right, pick D’s smallest digit (say it is b) Then E = acde^b Repeat. Examples: 23 --> 3^2 = 9 = stop 34 --> 4^3 = 64 --> 6^4 = 1296 --> 296^1 = 296 --> 96^2 = 9216 --> 926^1 = 926 --> 96^2 = 9216 = loop 35 --> 5^3 = 125 --> 25^1 = 25 --> 5^2 = 25 = loop 3252 --> 325^2 = 105625 --> 15625^0 = 1 = stop Which integers loop? Which integers stop? Best, É. (sorry if this has been explored before)
Since the vast majority of integers have a 0 digit, the vast majority of integers stop on the first step. Of the remainder, the vast majority will have a 1 digit, and so will enter a chain in which all the 1's get removed. Then, the vast majority will have a 2, and their lengths will double, and again the vast majority will have a 0 and stop. So I expect loops to be rare and get rarer. The stoppers dominate so aggressively that I expect that there will be either a finite set of loopers, or at best a small number of easily-recognized infinite classes of loopers, with a linear or quadratic upper bound on the number of loopers in each order of magnitude. 25 is the smallest looper. I would be interested to see any example of a five-digit looper. (Essentially I am hoping that somebody writes the code for me.) On Tue, Dec 29, 2020 at 12:20 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun,
Say D has 5 digits: D = abcde Starting from the right, pick D’s smallest digit (say it is b) Then E = acde^b Repeat. Examples: 23 --> 3^2 = 9 = stop 34 --> 4^3 = 64 --> 6^4 = 1296 --> 296^1 = 296 --> 96^2 = 9216 --> 926^1 = 926 --> 96^2 = 9216 = loop 35 --> 5^3 = 125 --> 25^1 = 25 --> 5^2 = 25 = loop 3252 --> 325^2 = 105625 --> 15625^0 = 1 = stop Which integers loop? Which integers stop? Best, É. (sorry if this has been explored before)
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If I haven't made a mistake these 31 integers are the only "reduced" loopers: 25, 26, 28, 34, 35, 36, 38, 43, 46, 52, 53, 62, 63, 64, 82, 83, 236, 239, 246, 254, 296, 326, 329, 362, 392, 426, 462, 524, 542, 926, 962 By reduced I mean they don't contain 1's. As you pointed out one can insert as many 1's as desired in any of these and still have a looper. This sequence is not in the OEIS. On Tue, Dec 29, 2020 at 8:06 PM Allan Wechsler <acwacw@gmail.com> wrote:
Since the vast majority of integers have a 0 digit, the vast majority of integers stop on the first step.
Of the remainder, the vast majority will have a 1 digit, and so will enter a chain in which all the 1's get removed. Then, the vast majority will have a 2, and their lengths will double, and again the vast majority will have a 0 and stop.
So I expect loops to be rare and get rarer. The stoppers dominate so aggressively that I expect that there will be either a finite set of loopers, or at best a small number of easily-recognized infinite classes of loopers, with a linear or quadratic upper bound on the number of loopers in each order of magnitude.
25 is the smallest looper. I would be interested to see any example of a five-digit looper. (Essentially I am hoping that somebody writes the code for me.)
On Tue, Dec 29, 2020 at 12:20 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun,
Say D has 5 digits: D = abcde Starting from the right, pick D’s smallest digit (say it is b) Then E = acde^b Repeat. Examples: 23 --> 3^2 = 9 = stop 34 --> 4^3 = 64 --> 6^4 = 1296 --> 296^1 = 296 --> 96^2 = 9216 --> 926^1 = 926 --> 96^2 = 9216 = loop 35 --> 5^3 = 125 --> 25^1 = 25 --> 5^2 = 25 = loop 3252 --> 325^2 = 105625 --> 15625^0 = 1 = stop Which integers loop? Which integers stop? Best, É. (sorry if this has been explored before)
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Compare: 52792-->5279^2; 52792-->5792^2; On Tue, Dec 29, 2020 at 11:49 PM Dan Asimov <asimov@msri.org> wrote:
Eric Angélini a écrit:
Starting from the right, pick D’s smallest digit
What does *where you start* have to do with this?
And what is the rule for when there is a tie?
(52792, for instance.)
—Dan
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I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent. I am curious to know how Edwin proved the completeness of his reduced looper set. I mean, how do you know there isn't some biggish number that gets squared or cubed a few times, until it's really gigantic, and then happens to be made mostly of 1's, which when removed leave the original biggish number? On Tue, Dec 29, 2020 at 11:49 PM Dan Asimov <asimov@msri.org> wrote:
Eric Angélini a écrit:
Starting from the right, pick D’s smallest digit
What does *where you start* have to do with this?
And what is the rule for when there is a tie?
(52792, for instance.)
—Dan
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Allan, you are right! It occurred to me after I pressed the send button that I was missing a proof. I only show that those I listed are the only reduced loopers up to 10^6. In particular there are no reduced loopers with 4, 5 or 6 digits. There could be one with 7 or more digits. --Edwin On Wed, Dec 30, 2020 at 12:01 AM Allan Wechsler <acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
I am curious to know how Edwin proved the completeness of his reduced looper set. I mean, how do you know there isn't some biggish number that gets squared or cubed a few times, until it's really gigantic, and then happens to be made mostly of 1's, which when removed leave the original biggish number?
On Tue, Dec 29, 2020 at 11:49 PM Dan Asimov <asimov@msri.org> wrote:
Eric Angélini a écrit:
Starting from the right, pick D’s smallest digit
What does *where you start* have to do with this?
And what is the rule for when there is a tie?
(52792, for instance.)
—Dan
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I suspect that this is one of those cases where we could produce a heuristic argument that strongly suggests that the "probability" of finding a larger looper is very small -- and we could make it smaller by extending the search higher. But I don't see any promising proof techniques. On Wed, Dec 30, 2020 at 12:17 AM W. Edwin Clark <wclark@mail.usf.edu> wrote:
Allan, you are right! It occurred to me after I pressed the send button that I was missing a proof. I only show that those I listed are the only reduced loopers up to 10^6. In particular there are no reduced loopers with 4, 5 or 6 digits. There could be one with 7 or more digits. --Edwin
On Wed, Dec 30, 2020 at 12:01 AM Allan Wechsler <acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
I am curious to know how Edwin proved the completeness of his reduced looper set. I mean, how do you know there isn't some biggish number that gets squared or cubed a few times, until it's really gigantic, and then happens to be made mostly of 1's, which when removed leave the original biggish number?
On Tue, Dec 29, 2020 at 11:49 PM Dan Asimov <asimov@msri.org> wrote:
Eric Angélini a écrit:
Starting from the right, pick D’s smallest digit
What does *where you start* have to do with this?
And what is the rule for when there is a tie?
(52792, for instance.)
—Dan
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Thanks, Allan. But I would hope Eric could write his rules unambiguously so that we don't need to surmise what they are from examples. —Dan
On Tuesday/29December/2020, at 9:01 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
25, 26, 28, 34, 35, 36, 38, 43, 46, 52, 53, 62, 63, 64, 82, 83, 236, 239, 246, 254, 296, 326, 329, 362, 392, 426, 462, 524, 542, 926, 962 By reduced I mean they don't contain 1's. As you pointed out one can insert as many 1's as desired in any of these and still have a looper. This sequence is not in the OEIS.
Ed, I hope you will submit 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 Wed, Dec 30, 2020 at 12:41 AM Dan Asimov <asimov@msri.org> wrote:
Thanks, Allan.
But I would hope Eric could write his rules unambiguously so that we don't need to surmise what they are from examples.
—Dan
On Tuesday/29December/2020, at 9:01 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
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OK, Neil. Will do. But I first want to improve my program and rerun it. --Edwin On Fri, Jan 1, 2021 at 8:37 AM Neil Sloane <njasloane@gmail.com> wrote:
25, 26, 28, 34, 35, 36, 38, 43, 46, 52, 53, 62, 63, 64, 82, 83, 236, 239, 246, 254, 296, 326, 329, 362, 392, 426, 462, 524, 542, 926, 962 By reduced I mean they don't contain 1's. As you pointed out one can insert as many 1's as desired in any of these and still have a looper. This sequence is not in the OEIS.
Ed, I hope you will submit 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 Wed, Dec 30, 2020 at 12:41 AM Dan Asimov <asimov@msri.org> wrote:
Thanks, Allan.
But I would hope Eric could write his rules unambiguously so that we don't need to surmise what they are from examples.
—Dan
On Tuesday/29December/2020, at 9:01 PM, Allan Wechsler < acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
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Le mer. 30 déc. 2020 à 00:44, W. Edwin Clark <wclark@mail.usf.edu> a écrit :
If I haven't made a mistake these 31 integers are the only "reduced" loopers: 25, 26, 28, 34, 35, 36, 38, 43, 46, 52, 53, 62, 63, 64, 82, 83, 236, 239, 246, 254, 296, 326, 329, 362, 392, 426, 462, 524, 542, 926, 962 By reduced I mean they don't contain 1's.
I get the same with (PARI) orbit(n, U=[n], m)={ while(n>9, my( m=vecmin(n=digits(n))); forstep( i=#n,1,-1, if( n[i]==m, n=fromdigits(n[^i])^m; break)); setsearch(U,n) && break; U=setunion(U,[n])); U} for(n=1,99999,vecmin(digits(n))>1 && (o=orbit(n))[1]>9 && printf("O(%d)=%d, ",n,o)) /* see below for output */ We see that if they are classified by m(n) := max( O(n)), the largest element of their orbit, there are only 8 classes, the largest being that with m(n) = 9216: O(25)=[25], O(52)=[25,52], O(35)=[25,35,125], O(53)=[25,53,125], O(26)=[26,36,216], O(36)=[26,36,216], O(62)=[26,36,62,216], O(63)=[26,36,63,216], O(38)=[25,38,52,512], O(83)=[25,52,83,512], O(239)=[25,52,152,239,1521], O(329)=[25,52,152,329,1521], O(392)=[25,52,152,392,1521], O(246)=[26,36,216,246,2116], O(426)=[26,36,216,426,2116], O(462)=[26,36,216,462,2116], O(28)=[28,64,296,926,1296,9216], O(34)=[34,64,296,926,1296,9216], O(43)=[43,64,296,926,1296,9216], O(46)=[46,296,926,1296,9216], O(64)=[64,296,926,1296,9216], O(82)=[64,82,296,926,1296,9216], O(236)=[236,296,926,1296,9216], O(254)=[254,296,926,2916,9216], O(296)=[296,926,9216], O(326)=[296,326,926,1296,9216], O(362)=[296,362,926,1296,9216], O(524)=[296,524,926,2916,9216], O(542)=[296,542,926,2916,9216], O(926)=[926,9216], O(962)=[926,962,9216]. - Maximilian As you pointed out one can insert
as many 1's as desired in any of these and still have a looper. This sequence is not in the OEIS.
On Tue, Dec 29, 2020 at 8:06 PM Allan Wechsler <acwacw@gmail.com> wrote:
Since the vast majority of integers have a 0 digit, the vast majority of integers stop on the first step.
Of the remainder, the vast majority will have a 1 digit, and so will enter a chain in which all the 1's get removed. Then, the vast majority will have a 2, and their lengths will double, and again the vast majority will have a 0 and stop.
So I expect loops to be rare and get rarer. The stoppers dominate so aggressively that I expect that there will be either a finite set of loopers, or at best a small number of easily-recognized infinite classes of loopers, with a linear or quadratic upper bound on the number of loopers in each order of magnitude.
25 is the smallest looper. I would be interested to see any example of a five-digit looper. (Essentially I am hoping that somebody writes the code for me.)
On Tue, Dec 29, 2020 at 12:20 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun,
Say D has 5 digits: D = abcde Starting from the right, pick D’s smallest digit (say it is b) Then E = acde^b Repeat. Examples: 23 --> 3^2 = 9 = stop 34 --> 4^3 = 64 --> 6^4 = 1296 --> 296^1 = 296 --> 96^2 = 9216 --> 926^1 = 926 --> 96^2 = 9216 = loop 35 --> 5^3 = 125 --> 25^1 = 25 --> 5^2 = 25 = loop 3252 --> 325^2 = 105625 --> 15625^0 = 1 = stop Which integers loop? Which integers stop? Best, É. (sorry if this has been explored before)
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participants (6)
-
Allan Wechsler -
Dan Asimov -
M.F.H. -
Neil Sloane -
W. Edwin Clark -
Éric Angelini