cp's OEIS Frontend

A329196 Irregular table whose rows are the nontrivial cycles of the ghost iteration A329200, ordered by increasing smallest member, always listed first.

%I A329196 #66 Jan 01 2024 22:47:16
%S A329196 10891,12709,11130,11107,11090,43600,44960,45496,44343,44232,44021,
%T A329196 74780,78098,76207,75800,78180,79958,77915,78199,79979,82001,110891,
%U A329196 112709,111130,111107,111090,180164,258316,224791,227119,232727,221172,220107,217990,201781
%N A329196 Irregular table whose rows are the nontrivial cycles of the ghost iteration A329200, ordered by increasing smallest member, always listed first.
%C A329196 A329200 consists of adding the number whose digits are the absoute values of differences of adjacent digits of n in case it is even, or subtracting it if it is odd. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence lists these cycles, ordered by their smallest member which is always listed first. Sequence A329197 gives the row lengths.
%C A329196 Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the (cyclic) first differences are also the same and add again up to 0.) This is the case for rows 1, 2, 3, ... (but not row 4 or 6) of this table. Rows 5, 7 and 8 are the second members of these three families. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.
%H A329196 Lars Blomberg, <a href="/A329196/b329196.txt">Table of n, a(n) for n = 1..235 (cycles found for n < 10^10)</a>
%e A329196 The table starts:
%e A329196    n |  cycle #n  (length = A329197(n))
%e A329196   ---+-----------------------------------------------------------------------
%e A329196    1 |  10891,  12709,  11130,  11107,  11090
%e A329196    2 |  43600,  44960,  45496,  44343,  44232,  44021
%e A329196    3 |  74780,  78098,  76207
%e A329196    4 |  75800,  78180,  79958,  77915,  78199,  79979, 82001
%e A329196    5 | 110891, 112709, 111130, 111107, 111090
%e A329196    6 | 180164, 258316, 224791, 227119, 232727, 221172, 220107, 217990, 201781
%e A329196    7 | 443600, 444960, 445496, 444343, 444232, 444021
%e A329196    8 | 774780, 778098, 776207
%e A329196    9 | 858699, 891929, 873052
%e A329196   10 | 1110891, 1112709, 1111130, 1111107, 1111090
%e A329196   11 | 3270071, 3427147, 3301514
%e A329196   12 | 4381182, 4538258, 4412625
%e A329196   13 | 4443600, 4444960, 4445496, 4444343, 4444232, 4444021
%e A329196   14 | 5492293, 5649369, 5523736
%e A329196   15 | 7774780, 7778098, 7776207
%e A329196   16 | 8858699, 8891929, 8873052
%e A329196   17 | 11110891, 11112709, 11111130, 11111107, 11111090
%e A329196   18 | 33270071, 33427147, 33301514
%e A329196   19 | 44381182, 44538258, 44412625
%e A329196   20 | 44443600, 44444960, 44445496, 44444343, 44444232, 44444021
%e A329196   21 | 55492293, 55649369, 55523736
%e A329196   22 | 77774780, 77778098, 77776207
%e A329196   23 | 85869922, 89192992, 87305285
%e A329196   24 | 88858699, 88891929, 88873052
%e A329196   25 | 111110891, 111112709, 111111130, 111111107, 111111090
%e A329196   26 | 333270071, 333427147, 333301514
%e A329196   27 | 444381182, 444538258, 444412625
%e A329196   28 | 444443600, 444444960, 444445496, 444444343, 444444232, 444444021
%e A329196   29 | 555492293, 555649369, 555523736
%e A329196   30 | 615930235, 670393447, 653027344, 665352754, 664129233, 666446943,
%e A329196      | 666244592, 665824445, 664462444, 666486644, 666728664, 666884866,
%e A329196      | 667089286, 668871048, 670887192, 653085505, 640702450
%e A329196   31 | 777774780, 777778098, 777776207
%e A329196   32 | 809513051, 898955405, 887815260, 888989606, 889100972, 887290047,
%e A329196      | 885711004, 888971108, 889097126, 891089740, 909270974
%e A329196   33 | 858699257, 891929989, 873052978
%e A329196   34 | 885869922, 889192992, 887305285
%e A329196   35 | 888858699, 888891929, 888873052
%e A329196   36 | 1111110891, 1111112709, 1111111130, 1111111107, 1111111090
%e A329196   37 | 3333270071, 3333427147, 3333301514
%e A329196   38 | 4444381182, 4444538258, 4444412625
%e A329196   39 | 4444443600, 4444444960, 4444445496, 4444444343, 4444444232, 4444444021
%e A329196   40 | 5461740619, 5587375277, 5618817627, 5461741482, 5587374828, 5618818294
%e A329196   41 | 5555492293, 5555649369, 5555523736
%e A329196   42 | 6615930235, 6670393447, 6653027344, 6665352754, 6664129233,
%e A329196      | 6666446943, 6666244592, 6665824445, 6664462444, 6666486644,
%e A329196      | 6666728664, 6666884866,
%e A329196      | 6667089286, 6668871048, 6670887192, 6653085505, 6640702450
%e A329196   43 | 7777774780, 7777778098, 7777776207
%e A329196   44 | 8858699257, 8891929989, 8873052978
%e A329196   45 | 8885869922, 8889192992, 8887305285
%e A329196   46 | 8888858699, 8888891929, 8888873052
%e A329196   47 | 11111110891, 11111112709, 11111111130, 11111111107, 11111111090
%e A329196   48 | 31128941171, 33145094237, 33376689451, 33417710965, 33281649034,
%e A329196      | 33114123103, 32910811890
%e A329196   49 | 44444443600, 44444444960, 44444445496, 44444444343,
%e A329196      | 44444444232, 44444444021
%e A329196 The smallest starting value for which the trajectory under A329200 does not end in a fixed point is n = 8059: This leads into a cycle of length 5, 11090 -> 10891 -> 12709 -> 11130 -> 11107 -> 11090. "Rotated" as to start with the smallest member, this yields the first row of this table, (10891, 12709, 11130, 11107, 11090).
%e A329196 Starting value n = 37908 leads after two steps into the next cycle (44232, 44021, 43600, 44960, 45496, 44343), of length 6. Again "rotating" this list as to start with the smallest member, it yields the second row of this table.
%e A329196 Starting value n = 68060 leads after 8 steps into a new cycle of length 7, (75800, 78180, 79958, 77915, 78199, 79979, 82001). However, this will NOT give row 3 but only row 4, because:
%e A329196 The starting value 70502 leads after 3 steps into the loop (74780, 78098, 76207) which comes lexicographically earlier than the previously mentioned cycle of length 7. Therefore this is row 3 of this table.
%e A329196 Starting value 70515 enters the loop (111090, 110891, 112709, 111130, 111107) after 15 steps. This becomes row 5.
%e A329196 This row 5 is the same as row 1 with the initial digit 1 duplicated in each term: it is the second member of this infinite family of cycles of length 5. Similarly, rows 2 and 3 (where all terms have the same length and initial digit) also lead to infinite families of cycles by successively duplicating the initial digit of each term.
%e A329196 The pattern 858699257(257|857)*84302(302|342)* also yields cycles. - _Lars Blomberg_, Nov 15 2019
%o A329196 (PARI)
%o A329196 T(n,T=[n])={while(!setsearch(Set(T),n=A329200(n)), T=concat(T,n));T} /* trajectory; is a cycle when n is a member of it */
%o A329196 {U=0; M=[]; for(n=9,oo, bittest(U>>=1,0) && next; if(M && n>M[1], print(T(M[1])); M=M[^1]); t=n; V=U; while( !bittest(U,-n+t=A329200(t)), t>n || next(2); U+=1<<(t-n)); bittest(V,t-n) || #Set(digits(t))==1 || M=setunion(M,[vecmin(T(t))]) )}
%Y A329196 Cf. A329197 (row lengths), A329200, A329198.
%Y A329196 Cf. A329342 (analog for the variant A329201).
%K A329196 nonn,tabf
%O A329196 1,1
%A A329196 _M. F. Hasler_, Nov 10 2019
%E A329196 Rows 9 through 35 from _Scott R. Shannon_, Nov 12 2019
%E A329196 Table of cycles extended by _Lars Blomberg_, Nov 15 2019