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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