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A329196 Irregular table whose rows are the nontrivial cycles of the ghost iteration A329200, ordered by increasing smallest member, always listed first. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

EXAMPLE

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

PROG

(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))]) )}

CROSSREFS

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

Cf. A329342 (analog for the variant A329201).

Sequence in context: A214243 A250896 A164824 * A083513 A084277 A307873

Adjacent sequences:  A329193 A329194 A329195 * A329197 A329198 A329199

KEYWORD

nonn,more,tabf

AUTHOR

M. F. Hasler, Nov 10 2019

EXTENSIONS

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

Table of cycles extended by Lars Blomberg, Nov 15 2019

STATUS

approved

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Last modified June 6 13:01 EDT 2020. Contains 334827 sequences. (Running on oeis4.)