login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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
KEYWORD
nonn,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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 14:33 EDT 2024. Contains 371254 sequences. (Running on oeis4.)