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

10891, 12709, 11130, 11107, 11090, 43600, 44960, 45496, 44343, 44232, 44021, 74780, 78098, 76207, 75800, 78180, 79958, 77915, 78199, 79979, 82001, 110891, 112709, 111130, 111107, 111090, 180164, 258316, 224791, 227119, 232727, 221172, 220107, 217990, 201781

Offset: 1

Author

M. F. Hasler, Nov 10 2019

Keywords

• nonn
• tabf

Comments

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.

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.

Examples

			The table starts:
   n |  cycle #n  (length = A329197(n))
  ---+-----------------------------------------------------------------------
   1 |  10891,  12709,  11130,  11107,  11090
   2 |  43600,  44960,  45496,  44343,  44232,  44021
   3 |  74780,  78098,  76207
   4 |  75800,  78180,  79958,  77915,  78199,  79979, 82001
   5 | 110891, 112709, 111130, 111107, 111090
   6 | 180164, 258316, 224791, 227119, 232727, 221172, 220107, 217990, 201781
   7 | 443600, 444960, 445496, 444343, 444232, 444021
   8 | 774780, 778098, 776207
   9 | 858699, 891929, 873052
  10 | 1110891, 1112709, 1111130, 1111107, 1111090
  11 | 3270071, 3427147, 3301514
  12 | 4381182, 4538258, 4412625
  13 | 4443600, 4444960, 4445496, 4444343, 4444232, 4444021
  14 | 5492293, 5649369, 5523736
  15 | 7774780, 7778098, 7776207
  16 | 8858699, 8891929, 8873052
  17 | 11110891, 11112709, 11111130, 11111107, 11111090
  18 | 33270071, 33427147, 33301514
  19 | 44381182, 44538258, 44412625
  20 | 44443600, 44444960, 44445496, 44444343, 44444232, 44444021
  21 | 55492293, 55649369, 55523736
  22 | 77774780, 77778098, 77776207
  23 | 85869922, 89192992, 87305285
  24 | 88858699, 88891929, 88873052
  25 | 111110891, 111112709, 111111130, 111111107, 111111090
  26 | 333270071, 333427147, 333301514
  27 | 444381182, 444538258, 444412625
  28 | 444443600, 444444960, 444445496, 444444343, 444444232, 444444021
  29 | 555492293, 555649369, 555523736
  30 | 615930235, 670393447, 653027344, 665352754, 664129233, 666446943,
     | 666244592, 665824445, 664462444, 666486644, 666728664, 666884866,
     | 667089286, 668871048, 670887192, 653085505, 640702450
  31 | 777774780, 777778098, 777776207
  32 | 809513051, 898955405, 887815260, 888989606, 889100972, 887290047,
     | 885711004, 888971108, 889097126, 891089740, 909270974
  33 | 858699257, 891929989, 873052978
  34 | 885869922, 889192992, 887305285
  35 | 888858699, 888891929, 888873052
  36 | 1111110891, 1111112709, 1111111130, 1111111107, 1111111090
  37 | 3333270071, 3333427147, 3333301514
  38 | 4444381182, 4444538258, 4444412625
  39 | 4444443600, 4444444960, 4444445496, 4444444343, 4444444232, 4444444021
  40 | 5461740619, 5587375277, 5618817627, 5461741482, 5587374828, 5618818294
  41 | 5555492293, 5555649369, 5555523736
  42 | 6615930235, 6670393447, 6653027344, 6665352754, 6664129233,
     | 6666446943, 6666244592, 6665824445, 6664462444, 6666486644,
     | 6666728664, 6666884866,
     | 6667089286, 6668871048, 6670887192, 6653085505, 6640702450
  43 | 7777774780, 7777778098, 7777776207
  44 | 8858699257, 8891929989, 8873052978
  45 | 8885869922, 8889192992, 8887305285
  46 | 8888858699, 8888891929, 8888873052
  47 | 11111110891, 11111112709, 11111111130, 11111111107, 11111111090
  48 | 31128941171, 33145094237, 33376689451, 33417710965, 33281649034,
     | 33114123103, 32910811890
  49 | 44444443600, 44444444960, 44444445496, 44444444343,
     | 44444444232, 44444444021
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).
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.
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:
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.
Starting value 70515 enters the loop (111090, 110891, 112709, 111130, 111107) after 15 steps. This becomes row 5.
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.
The pattern 858699257(257|857)*84302(302|342)* also yields cycles. - _Lars Blomberg_, Nov 15 2019
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..235 (cycles found for n < 10^10)

Crossrefs

Cf. A329197 (row lengths), A329200, A329198.

Cf. A329342 (analog for the variant A329201).

Programs

PARI

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 */
{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))]) )}

Extensions

Rows 9 through 35 from Scott R. Shannon, Nov 12 2019

Table of cycles extended by Lars Blomberg, Nov 15 2019